A MATHEMATICAL  DETERMINATION  OF  THE 
MAXIMUM  PRESSURE  AND  THE  EXTENT  OF 
COMBUSTION  IN  THE  GAS  ENGINE 

BY 

GEORGE  THEODORE  FELBECK 
B.  S.  University  of  Illinois 
1919 


THESIS 


Submitted  in  Partial  Fulfillment  of  the  Requirements  for  the 

Degree  of 


MASTER  OF  SCIENCE 
IN 

MECHANICAL  ENGINEERING 

IN 

THE  GRADUATE  SCHOOL 

OF  THE 

UNIVERSITY  OF  ILLINOIS 


1921 


Digitized  by  the  Internet  Archive 
in  2016 


https://archive.org/detaiis/mathematicaideteOOfelb 


*Required  for  doctor’s  degree  but  not  for  master’i 


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T:‘JBLS  OF  CONTENTS 


Page 

I.  Introduction.  1 

II.  Derivation  of  Equation  for  Adiabatic  Compression.  2 

III.  Derivation  of  Equilibrium  Equation  for  Constant 
Volume  Adiabatic  Combustion  involving  only 
the  Reaction  CO  f'2<)^->C0j_  Case  I.  6 

IV.  Derivation  of  Energy  Equation  for  Case  I.  15 

V.  Numerical  Example  Case  I.  17 

VI.  Derivation  of  Equilibrium  Equations  for  Constant 
Volume  Adiabatic  Combustion  involving  the 
Reactions  C 0 -t-iOj^-tCO^  and  CO^-t  CO  + -|0  Case  II.  19 

VII.  Derivation  of  Energy  Equation  for  Case  II.  24 

VIII.  Numerical  Example  Case  II.  24 

IX.  Derivation  ofEquilibrium  Equations  for  Con- 
stant Volume  Adiabatic  Combustion  involving 
the  burning  of  CO,  and  C and  the  Dis- 
sociation of  C 0j_  and  H^^O  ^ Case  III.  27 

X.  Derivation  of  Energy  Equation  for  Case  III.  24 

XI.  Redustion  of  Equations  and  Method  of  Solution, 

Case  III,  27 

XII.  Numerical  Example  Case  III,  41, 

XIII.  Effect  of  loss  of  Heat  during  Combustion  on  the 
Maximum  Temperature  and  Pressure  and  the  Ex- 
tent of  Combustion  in  Case  III,  45 

XIV.  Effect  of  Excess  Air  on  the  Maximum  Temperature 

and  the  Extent  of  Combustion  in  Case  III,  50 

XV.  Extension  of  Analysis  to  Cover  G&seous  Combus- 
tions Involving  the  Hydrocarbons  C^H^  0JL,Cz^yj 
and  C ^ H^  . 54 

Appendix  A,  Specific  Heats.  56 

Appendix  B , Heating  Values.  88 

Appendix  C , Chemical  Equilibrium.  122 

Appendix  D,  Determination  of  the  Maximum 

Possible  Percentage  of  N 0 
Pr4sent  at  the  Point  of  Maxi- 
mum Temperj:ture  in  the  Gas 
Engine . 144 


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4 


LIST  OF  CURVE  SHEETS 


Page 
19  a 


Gr^hical  Solution,  Case  I 


5 

TT 

TT 

" II 

26 

6 

n 

TT 

” III 

44 

7 

TT 

" with  loss  of  heat,  Case  III 

48 

7 (a) 

Curves  showing 
combustion  on 

effect  of  loss  of  heat  during 
maximum  explosion  conditions 

49 

8 

Graphical  solution  with  excess  air.  Case  III 

52 

8 (a) 

Curves  showing  effect  of  excess  air  on 
maximum  explosion  conditions. 

53 

9 

Specific  heat 

curves 

for  C Ox. 

61 

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TT 

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77 

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” nitrogen 

79 

18 

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IT 

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82 

19 

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TT 

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86 

20 

Dissociation  of  G 0^ 

128 

21 

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the  Reaction  i-C  0 

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1 


A Mathematical  Determination  of  the 
Maximum  Pressure  and  the  Extent  of  Oomhustion 
in  the  Gas  Engine 

I.  Introduction. 

The  maximum  pressure  obtained  in  the  cylinder  of 
a gas  engine  as  measured  from  indicator  cards  is  much  less  than 
one  is  led  to  expect  from  computations  using  the  "air  standard" 
cycle  with  air  as  a medium  having  constant  specific  heat.  In 
fact,  the  actual  pressures  are  about  one-half  those  obtained  by 
calculation.  Some  authorities  claim  that  this  discrepancy  is 
due  to  the  fact  that  the  specific  heat  of  the  actual  medium 
used  in  the  gas  engine  increases  with  the  temperature.  Others 
maintain  that  combustion  is  incomplete  and  thus  the  pressure  is 
lower.  Some  say  that  both  causes  contribute  to  the  decrease 
in  pressure. 

To  foretell  with  any  degree  of  accuracy  from  theo- 
retical considerations  what  the  final  result  from  a gaseous  com- 
bustion will  be  the  physical  and  chemical  properties  of  the  gase  s 
involved  must  be  studied  and  not  the  properties  of  some  other 
gas  entirely  foreign  to  the  combustion  under  consideration.  aVhiL  e 
air  is  one  of  the  main  constituents  involved  in  the  gas  engine 
combustion  process,  the  properties  of  the  gaseous  mixture  are 
quite  different  from  those  of  air.  The  gaseous  mixture  varies 
in  different  engines  with  the  proportion  of  air  to  gas  and  with 


2. 

the  nature  of  the  gas  itself  whether  natural  gas,  producer  gas, 
coal  gas,  or  any  of  the  other  suitable  gases.  The  physical 
and  chemical  properties  of  several  different  gaseous  mixtures 
will  in  general  be  different  although  they  may  all  have  the 
same  constituents.  The  properties  of  all  the  constituents  are 
not  the  same  so  that  v/hen  the  proportions  making  up  the  gaseous 
mixture  are  changed  the  properties  of  the  mixture  as  a whole 
are  changed.  It  is  not  reasonable  to  suppose  that  all  gase- 
ous mixtures  when  burned  under  the  same  conditions  as  to  initial 
pressure  and  temperature  will  all  produce  the  same  maximum  pres- 
sure. Each  case  is  a special  one  and  no  blanket  calculation 
can  be  made  to  cover  them  all. 

It  is  not  fair,  moreover,  to  an  engine  to  measure 
its  performance  in  terms  of  an  ideal  engine  using  a distinctly 
different  medium  such  as  air.  To  be  able  to  judge  the  state 
of  perfection  of  engine  design  the  ideal  established  must  be 
that  which  represents  the  maximum  results  obtainable  with  the 
given  working  medium. 

II.  Derivation  of  Energy  Equation  for  Adiabatic  Compression. 

In  this  discussion  only  two  phases  of  the  Otto 

(2) 

cycle  will  be  taken  up;  namely,  (1)  compression  and/oombustion. 
Both  processes  will  be  considered  adiabatic.  At  the  beginning 
of  compression  the  pressure  will  be  taken  as  14.7  lb.  per  square 
inch  and  the  temperature  as  190°F.  Hopkinson*has  shown  that 
the  temperature  of  the  charge  at  the  beginning  of  compression 
is  in  the  neighborhood  of  200°P.  The  value  190®  has  been 
chosen  arbitrarily  for  convenience.  It  will  be  assumed  that 
*Proo.  Inst.  M.E.  1908,  Vol.  Jan.  to  May. 


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3 


ignition  takes  place  at  the  end.  of  compression  and  that  the 
combustion  is  instantaneous.  The  combustion  will  thus  occur 
at  constant  volume.  It  will  be  assumed  that  the  constituents 
of  the  gaseous  mixture  obey  the  perfect  gas  law  PV/T  “ constant.* 
The  following  notation  will  be  used: 

State  1 (beginning  of  compression)  P, , V,  , T, 

State  2 (end  of  compression)  P;^ , , T^j^ 

State  3 (point  of  maximum  pressure)  P , ^ 

P=lbs#  per  sq.  ft.  V -cu.ft.  T = absolute ‘’P. 

The  compression  ratio  V,  /V^  is  determined  by  the 
design  of  the  engine.  Knowing  the  compression  ratio  and  with 
the  assumption  of  P,  - 2116  lbs. per  sq.ft,  and  T,^650  , the 
values  of  P^  and  T^  can  be  calculated  from  the  following: 

The  energy  equation 

dq  = du  + A pdv  (1) 

becomes  for  the  adiabatic  case 
du  -h  Apdv  =*  0.  (2) 

Also  we  have 

du  ■=  dt.  ( 3) 

Where  is  the  instantaneous  specific  heat  of  the  gas  mixture 
per  mol  at  constant  volume. 

The  specific  heat  of  a gas  can  be  accurately  rep- 
resented as  a function  of  the  temperature  for  the  range  of  tem- 
perature occurring  in  the  gas  engine.  This  function  is  of  the 

following  form: 

■=*  ^/77  't'  ^ 3 fp)  ~T  ^ (ft ) 

*Por  a discussion  of  the  accuracy  of  this  assumption,  see 
British  Association  Committee  Report,  P .368 , "Gas, Petrol  and 
Oil  Engines,"  by  Dugald  Clerk. 


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4. 

Combining  (2),  (3)  and  (4)  we  have 

The  perfect  gas  law  gives 
PV«M,RT  (6) 

where  M,is  the  number  of  mols  in  the  charge 

P„M,RT  (7) 

T" 

Substituting  (7)  in  (5)  and  dividing  by  T,  we  have 

o-‘ {-^  1- dT  + AM.Tf  ^ (a) 

Integrating  between  the  limits  (l)  and  (2) 

/Ofe  ^ -f  i tin  +^fm( K'-T.V  = A fl.Tf  /<»fe  ^ (V 

It  is  sometimes  more  convenient  to  use  the  pressure  relation 
P^/P,  than  the  volume  relation  V, /V^  . Prom  (6) 


Then 


V.  ^ T,.  Pi, 

■ T.  ^ 


(10) 


oo 


Substituting  (11)  in  (9)  and  collecting  terms  we  have 

(ani-  A n,l^)  /•JS'e  ^ t Z 6„(n-r,)  -t  f •=  A M,nl'=3e  ^ 0^) 

changing  to  common  logarithms  and  dividing  by  the  coefficient 
of  the  first  term  (12)  becomes 


Z b, 


m 


{Tt-T.) 


3 t 


m 


z.  fe-7;V 


_5_  (/3) 

A ^ r,  ‘ 


_ AM,  F 
A /I 

Transposing  all  known  quantities  to  the  right  hand  side  of  the 

3fr 


equation,  we  have 


23ozc(am-t  A/i,lf) 


71  -b 


i/B. 


2 • 2»302C(ar„  f Abf,P) 


_ _ AAf,/?  , /> 

<*n,fAM,R  ^ * A 


■b  -*■  ^ 


SL. 


3fr 


' ^ ! m T**-  fj^) 

Equation  (14)  can  best  be  solved  by  trial  for  the  unknown  T^* 


, 

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•V 

. /. 

ao;i? 

* « 

rt 

- ^ .T" 

,V 

sY-.i  ,®ii  ^ 1 !.: 

. - .;  Gti  J.  - . < . 

; -u  (I/! 

> ■''  ‘r»5^  K - ft 

1 . 

!>y  V 

»' 

V ‘ ■ > 

X.;.i  ; i ' - I > .. 

ii 

u.-  . r^^' 

\C.i.  ilC  * ■■‘l'‘■, 

* 

>1)  rii:’!- 

it<.:Zi  . to 

>'  ^ J 


^ 'i  S ,'» • I >*  ► ^ 

; - . t ^ r^ 


y 

<'.%  ^ 


r\ 


'r  m'' 


Lil;^A  . .'  ^ . ;■••;•.’  oi,  eoi . ...v rj  ..;  'rcAOt.  ' ' .' ^ 

PV:'  :■;■  Oi,'  , 



^A  K V .,  l^^  :.  ^ '><'  t V| 

^ '.^  . - . ) T fr. 

^ ■.v.,-v,v-'^  " ■ ,,^ 

i A '‘A.\,ir.^r  I.  .;  \.x  . .'.  .'  z^  i/cvioc  c > , ;t'-‘.‘  ( i*I ) 


,.r  '’  ■.v.,-v,v-'^ 


r^jtxsr-sssssraaei  ■ 


5. 

The  accuracy  of  the  value  of  derived  from  eq.ua- 
tion  (14)  depends  on  the  accuracy  of  the  value  tafcen  for  T^  and 
upon  the  accuracy  of  the  specific  heat  equation 
a t ZbT  -hSfr^ 


The  value  assumed  for  the  temperature  of  the 
charge  T may  he  accepted  as  accurate  within  20°  S’.  This 
temperature  varies  somewhat  with  the  load  but  as  the  value  190°F 
was  determined  at  full  load  and  this  treatment  deals  only  v/ith 
full  load  we  need  only  one  value  for  this  suction  temperature. 

The  equations  for  the  specific  heats  that  are 


available  and  that  may  be  accepted  as  accurate  for  the  range  of 


temperatures  encountered  in  the  gas  engine  are  as  follows: 
5’or  the  gases  GO,  0;^ , For  G^H^ 


= 4.51  + 0.5556  e 
Tp  * 6.5  + 0.5556  & 


7p  * 6.19+8.1  0 
yTy  ' 4.20+8.1  0 


For 

4.51  + 0.0005  O 
= 6.5  + 0.0005  0 


For  G^H-y 

rp>  • 6.67  +6.8  e 
■jr'y  * 4.68 -f-6. 8 0 


For  00^ 


For  G^H^ 


' 5.42 -^3.5  6-  0.48  a’’  }Tp^  6.43+11.3  0 

y-p-  7.41+3.5  0-  0.48  0^"  > 4.44  +11.3  0 


For  H^O 


For  G^H*  ( vapor) 


5.04  + 1.250  +0.2  0^  T’p  » 4.00  + 31.8  0 

7.03  +1.25  & +0.2  Ty  ^ 2.01  + 31.8  0 


For  G 

Yy-  4.04  + 5.0  0 
•y  = 6 . 03  +5.0  0 

Vp 

V/here  Yy  - instantaneous  specific  heat  per  mol  at  constant 
volume  and  © » where  T =r  degrees  Fahrenheit  absolute. 


./:TJ 


■.'j .i.  . ..  ^*T ' « fJi-'  ’ 


■ •_  -*  : -%J  t - i . * O *•  - rV  ^ 


'I 


'J  -■  ■ . -, 


-'  - ' . . • 

.l>i■i^  : '■T-w; 

L f ■:  -.i.i':.:  J'lfi  C 

' j .-i,- rr  ei 


‘ - .,  I-  ■. 

'»  ./  . ' i j.'.i.J.  aJ 


-I  1 

*•»*  !*■.  V ^ \ i’ 


( V 


. .'  c> 

. > 


. i , 


“ # * # 

: / 


■!  .•) 


r, 


> '•  i.<  J ; l.  i'  : 


.1  •■'..• 


;.;rlcv 


6« 

For  a discussion  of  specific  heats  see  Appendix  A. 

While  there  are  not  sufficient  specific-heat  data 
to  give  accurately  the  variation  of  specific  heat  with  temper- 
ature over  any  considerable  range  of  temperature  for  acetylene 
ethylene  Ethane  and  benzene  a 

large  error  in  the  specific  heat  equations  of  these  gases  intro- 
duces no  appreciable  error  in  the  calculation  of  the  compression 
temperature. 

Talce  for  instance  an  engine  running  on  acetylene 
(C^H^)  with  the  theoretical  required  air; 

Acetylene  burns  according  to  the  reaction 

^ /Vj  i-  -h  S /V^  ^ z COx.  -*•  9.  S'A/^ 

From  the  above  reaction  it  is  seen  that  the  acetylene  consti- 
tutes only  9.1^  by  volume  of  the  original  mixture.  If  the 
specific  heat  equation  we  use  for  acetylene  gives  results  that 
are  as  high  as  30^  in  error,  the  specific  heat  of  the  whole 
mixture  is  then  only  ^ in  error.  This  error  is  less  for  the 
other  hydrocarbons  mentioned  above  because  the  ratio  of  required 
air  to  gas  is  greater  than  for  acetylene.  It  is  doubtful  if 
any  of  the  specific  heat  equations  used  give  results  which  are 
correct  to  within 

Knowing  the  compression  pressure,  temperature,  and 
volume  we  have  the  initial  conditions  on  which  to  base  our  cal- 
culations for  the  combustion  phase. 

III.  Derivation  of  Equilibrium  Equation  for  Constant  Volume 
Adiabatic  Combustion  Involving  Only  the  Reaction 
CO  . Case  I. 


. K ' jJ..,:::, 


J i-> b V 


^ f 


>-  ■-  -.  V*  t' 

* -H*' 

- .'■ 

* > ». 

w '...(  . ’k  ..  s.'.. 'v>  L'.‘.  (Y 

~ 1 C.'  ,.  . .*•. 

*/ 

• ;■  ./  i 

1;  - 

X*  ■ 

ir  U 

' ..  t. , ..  s t» ^ a 1 V ..  Ci^  . V i.i  (;>•' 

■ T. . j.  ' «.  • 

<- 

*’  V*  T’ 

• *v 

'X  . 

r ; ; 

-'  ■ 

0 k'k  I ..  C MOTC  irx  ;, 

•'  • » 

r 

V i 

C *,-'•< 

:ir  ■' 

I 

. * / V.  i L ; 1.  ) 1 ...w  . ^ ( <^U  ) 

" i.  J ; 

Ob;?.  . 

. .. 

: .i  . '.w 

,J_ 

L i J..  m;..<  ;;I  *l.O\'C©  0' 

•V. -•■ 

'■'  0 

u . . \ 

'' 

.1  .U 

..  1 

t*.  •-  ' * . '^  t . t , 'v  i • C V X‘^, 

u-  Ti  ‘ 

i ' 

:r-  - k’. 

..r 

f 

■ ■ ' . 

1 

. X - V.;.  < 'lj  •',  o:’.i7 

* • wk.  - ' 

.-i  •;  !•  :’vi7  {^I: 

- 

*,  1* 

Lcr/ 

^ ..I  ■ w 1 ^ ..J  1^,  V-'  . V*i  1 t>-X^  V?  C-‘  f* 

I.  L: 

.:i/ ' 

' K- 

- 

.:.,  • 

V i .M  ^ V rv  j-*;  sfiJ 

* i.  \* 

'•  i 

• - u 

> «»• 

; .. 

j,  :'.i 

^ f.'i.  ft Y v;,-.o  .it.;;;.: 

U * K, 

. *• 

. J. 

r . • • " 

- ' 

« *ri 

’ J‘"  ‘ 

■ ' ■ 

u '':  f!  ^ 07?  ;r''i,?  . ;*;  o v.;i  xi.*i  :cffr/ 

i)  Oj  j 

I 

.'  r. 

i' 

r 1 :i 

CO 

L •;.:  :.;  '©-.  nr>ir(  ‘fe  ’ »o.. 

OfC  ■;•  : 

rf'  : 

' K '- 

- 

/:i  ‘ ci 

r)r.-  . J 'J  r ::  • ■. 

1C 

* '« 

**  e' 

r 

»'rC(fn  -M-.O'.  ■ •■•  ..  ,' 

L.rii 

•;;,  c' 

f • * ^ 

7 

• 

# 

. .'i 

Cw  t 

*1D1  l;.’  :2l  trS'^-C^ 

o'lj 

:.  V 

V • y 

■,'vi. 

, ^ r- 

i.C^^aw,  (511  Lo'.  k'.i  on.,-:  'I'C 

-X>3  0 V ! 


i.;  f>d;t  %J‘:< 


et  ' ,'  »•<  ■'  - ■ ^ 


I J 

iurfuiiR’  y..  .t;;t»'i'i;:,f  || 

1* 

••  . -;-  'r'^  ')!>  r-  1-^' «Vi  -"•  , 

t:-  ■/  6i  dc.ii.y*  .■(’  crfJ  ’' 

, ''j';"?  uOi  j'aycfj-ito  ‘to.  '^r?>  i j-^  Ok) 

^r'.;I.  V a Mv  fAii'- ’ . ■■(■/*<:  i;vAi.;rft:  .• '*  ? 'f'  /■'oO’-  vi':\-'^  .^11 


7. 

We  will  ta^e  first  the  case  of  an  engine  rimning 
on  a simple  gas  such  as  CO.  It  will  he  assumed  that  none  of 
the  products  of  combustion  of  the  previous  cycle  are  left  in 
thecylinder  so  that  the  initial  charge  consists  only  of  carbon 
monoxide  (CO),  oxygen  (0^),  and  nitrogen  (H^.) . 

Upon  ignition  of  the  charge  it  is  possible  for 
the  following  reactions  to  take  place;  namely, 

C0^i0^->C0^  (1)  U^i-O^-^ENO  (2) 

It  will  be  shown  later  that  reaction  (2)  does 
not  take  place  to  any  appreciable  extent  so  that  it  may  be  omit- 
ted from  the  discussion.  (See  Appendix  D.) 

Reaction  (1)  progresses  until  chemical  eq.uilibrium 
is  established  at  which  point  we  have  the  maximum  pressure  and 
temperature  attainable  in  the  given  case.  The  proportions  of 
the  various  gases  present  will  remain  the  same  until  some  change 
of  pressure  or  temperature  is  made.  This  is  what  occurs  when 
the  engine  piston  moves  forward.  The  gases  expand  doing  useful 
work  and  the  pressure  and  temperature  decrease.  At  ordinary 
temperatures  reaction  (1)  will  go  nearly  to  completion  while  at 
the  high  temperatures  attained  in  the  gas  engine  cylinder  consid- 
erable CO  and  0^  may  be  present.  As  the  temperature  decreases 
during  expansion  a greater  part  of  the  CO  present  burns  to  CO^ 
so  that  at  the  point  of  exhaust  the  reaction  is  practically  com- 
plete. 

When  a chemical  reaction  between  two  substances 
is  first  started  the  vigor  of  the  reaction  is  great.  Depending 


^ I 


. -to  J.\l.  ' J^xif  ;v 

: ouCi.  j-  ,j  ■<-■.•  . jt;;j',  Jr.  .0;  t.:  itiv.  a/2,  .'i  i..:j  /a 


.:  X .U-u  fT' ■ 

■ ..iI,ircX Col  erf;i 


(..) 

L‘.  ■ - 

V.  ' . It 

{ 

' 'i  / 

- 1 

^ cr 

• 1 ^ 
V ‘ V 

1 

i 

.. . -,  '■■■ 

♦ • . w . . *) 

• i.'  -•■  J- .:. 

0 

J ^ 

u . ^ ^ - 

w • ^ ' 

tiiJci ; w'l  J 

■l[.i  .!■  '.'u 

jC-rt 

• 

1 , 

.•J 

- 'ji  fc'D«r 

Ii>.y  0 

r .. 

.U  V. 

\ ...'C  / 

- ■’.*,  .;»  'i  ' 

-■■.  ■ ;•  ■ ( i. 

) K>iJ  t .'/.' 

1 ^ w-  *riirf  •>  -b  . 

I 

.‘x'r.Z  :i  ? 

£■  ;V*  .:  . 

’ . .' 

iii'iMr.'  J , 

:..r^ftiX.fbcfe;6  <i.f 

- -*■  ^«IZ),UO 

crj? 

i 1 j njv 

i;  fi'i.,  i.: 

f)I.;  . . i y- 

^ .rr*?  X 

£Lf 

-1..  ill 

, ;..  .. 

■•  ■••  b;jR;^3  -,jjr  »-■  v,'  »nt 

^ , 4 ' • ■ . ' 

V 

*.l- 

1\'  . . 

...  . r f 1 

to  f)\.';-;Ob'j;fr  iyt 

e.ffj'  Bff'  t 

riC  iu  4 :?)T.  i6‘ii.  r<tr  "^ 

'•i'&'T/oj  L-f?u 

rO  rjfwi  C?  ■ Eo'!t  :' 

V , 

i,  ’io  *'i  . i/ir^  i\  n-ple'rLsqjtf  . l^tysrl' 


: *r -./i -i.'i>-'  . .-.i-,  j> *•.•_. <.".:•*  'r'c.-,i:'  .c 

£ C ■:>  . Cu  /i  ■:,  . ' •-,  ^ / (£} 

ilvC  O . :'C  i' J .’i  r>{iC  f.  f 


~ i J.  , I><.'.i' . ' 


^-1.. 


8 


on  conditions,  the  reaction  tends  Id  go  one  way  or  the  other. 

This  tendency  of  the  reaction  to  proceed  we  may  call  the  driving 

force  of  the  reaction.  As  the  reaction  proceeds  this  driving 

force  decreases  and  the  reaction  slows  up.  YiHaen  chemical  eo[uil- 

ihrium  is  established  this  driving  force  has  been  reduced  to 

zero  for  there  is  no  tendency  for  the  reaction  to  proceed  in 

either  direction.  If  a reaction  should  acquire  considerable 

momentum  and  proceed  past  the  equilibrium  point,  the  driving 

force  becomes  negative  and  the  reaction  is  reversed  and  proceeds 

in  the  reversed  direction  until  equilibrium  is  established. 

c 


We  may  represent  a 
chemical  reaction  by  fig.l.  The 
point  a represents  the  constituents 
in  their  initial  state.  Point  b 
represents  the  reaction  completed. 
TaJcing  the  reaction 
00^ 


at  point  a we  have  only  CO  and  0;^,  while  at  point  b there  is 
only  CO^.  The  progress  of  the  reaction  is  indicated  by  x. 

The  ordinate  of  curve  n is  the  driving  force  A of  the  reaction. 
The  amount  of  mechanical  work  that  can  be  done  by  the  reaction 
up  to  any  point  P is  represented  by  the  area  between  curve  n and 
the  axis  a,b  between  the  limits  a and  x.  This  can  be  repre- 
sented by  the  equation 

aW;,  = = E (is) 


9. 


Differentiating  eqi.(l5) 
dW  =Adx  ’■dS 


A 

dx  ~ 


(16) 


From  equation  (16)  the  equilibrium  point  can  be 
determined  by  putting  A^O,  This  equation  involves  the  varia- 

ble S and  X.  We  may  proceed  as  follows  to  express  E in  terms 
of  X* 

Since  the  laws  of  thermodynamics  are  applicable  to 
chemical  reactions  we  have  the  following: 
dQ  ® du  i-dW  «du-l-dE.  (17) 
dQ  - heat  from  external  sources 
du=  change  in  internal  energy 
dW  e external  work  done  dE. 

Also  for  a reversible  process, 
dQ=TdS  (18) 

Combining  (17)  and  (18), 

dE  ^TdS  - du  (19) 

Substituting  (19)  in  (16)  we  have 

T ds  du  , , 

dx  ~ dx  ^ 

The  variables  S and  u can  be  expressed  in  terms  of  T and  x so 
that  at  a given  temperature  the  value  of  x at  the  equilibrium 
point  can  be  determined  by  putting  A “0  in  equation  (20).  To 
transfer  equation  (20)  into  one  involving  x and  T as  variables 
we  proceed  as  follows: 

For  the  reaction  under  consideration;  namely, 

cot 


(a£) 


{•t^  - 

-Si  »,  t ' C *i  ^ 

• f'-  e'  ' 

•.. ..  v"‘>‘  ^ ■' 


. ur 

. L2U^  h blu 
. to 


V L ■ • " t‘  ■ '.‘.‘i  u 

:©rX 

' 7£  ; . .'  ^ u;:  » -o 

pi) 

r f r r •»  jjr  .1  ^ r 

aL'C'xi  J.’trd  - i i' 

v:rvv,  A-..;!.*  t 

■ •.Jt  ajm»idQ )/0 

. .ij  V . t;  :,;•!(  ri7 

I' .<;’^0v^^'^  - . > 

, . V '..  N ■>  ‘-  i. 'J'^^'V 

*V  0 ^ X^'- 

■ {o,n 

^^5T  ii 

*s  I . * W KJ  •_'  V*  *<  A 

I * 

1'  Tv  c>>'X''V 

C'CliC  ; :f:IJj‘r 

^ v .‘  .'3i  ^r; 


, -Mi  l 

. 't 

) Xfv 

. (V.:) 

0 

(V.:  )■•';: 

.,  jji}  '•  -Xii' 
<•.  - • . 

4.  C . 

• { 'Xr: 

•.  : 1 

.X)  ■■,.,,..i,;JX.r5vi 

k ‘.J  Nat 

iiSJ 

'^'  XV 

XvV-  X 

• • C f 

liiT ' 

t ir.  y I 

:i  ‘ '.'.  .v.:v  e 

4 ^ •* 

k 

/*»’«*"  t f ♦ A 1 

M»  * ^ W •)  . li*  k.'  ’ 

1,  L . V,’  y 

.:,  X 1 'X. 

^iUi 

' ^ . 

t/ 

H*  ^ ’ * • 

</  Oij  V J 

vA*i 

Oq 

cx..;  (p' 

«•’  'fv>X.7ri;: 

: : 0 i!  J.o't  as  y c » cj  . iv 

■■••.,:.  /■ 

■ r , 

^ L CtP  -V’  it  J jlfv  i l5-i>  X *.*  0 *0  V ^ C J ^ V*  '&  ^ ^ 

O-iV  00 


10. 


the  initial  mixture  is  as  follows  using  an  e xcess  of  air, 


CO 

0* 


Mols 

1 

e 

f 


sum  m 

The  gas  mixture  at  any  point  x in  the  progress 

of  the  comhustion  is  as  follows: 

Mols 
GO*.  X 
GO  (1-x) 

0.-  (e-^) 

.1  , 

sum  m - isic  m 

Let  P he  the  pressure  of  the  gas  mixture  in  the 
state  x;  then  the  partial  pressures  of  the  oonsituents  are 


p --P 


CO 


1-x 

m' 


= I.P 


(£1) 


The  energy  U of  the  mixture  is  the  sum  of  the  en- 
ergies of  the  constituents;  that  is, 

U =XU  (l-x)u  f fu 

m 

differentiating  and  multiplying  by  -1 


iS. 

dx 


CO 


+ iu  - U 


c<J, 


(££) 


This  expression  is  simply  13ie  heat  of  combustion 
H V at  constant  volume  and  at  the  given  constant  temperature  T. 

varies  with  the  temperature.  Since  y-  A 
the  energy  U is  a point  function  it  is 
independent  of  the  path  so  that  the 
same  amount  of  heat  is  liberated  in 

Pig.  2 

going  direct  from  A to  D as  is  evolved  by  taking  the  circuitous 


I 


. J. 


f ' <*  *■*  ^ 4 .«»■  * . (X  > V V 1^-  •€>.  ,,  - -X 

•i.,j  . ii.: 

V X Xi  i 

cXoM 

X 

UO 

X '■  ^ 

^ ^ _ --  J, 

u 

ii 

• i ' » 1 * ’ ^ ■ • ‘ . • ■*  • >•  - . t , . 

s^  ^ * \ ^ “*  44  *■  * *'  ■*'  ' ' t i *.  '4  i.,  ^ ‘ ^• 

i-hA' 

* O'  0 jw  J.  c>  J.  '<i  jL 

! 

a i-JiSwC^'v 

c urfd 

- 

«ro-.f 

^crx 

■< 

' ”1) 

r*i  /• 

V 

. ' r V 

(^  yT') 

0 

“ .-,  ■ ,'  ■ ' ' ■"-■■■>  : ' 

•! 

X 

fr  - nr"  i 

' ‘ I*  • 

i J vj.,  ‘.-I-  ■».•  r,, ' ''  ^ s..  J 'J  > i,  ’ ‘ 


, ;i  J".,-  ; c rio,Liv  i J4r'.vi>  e.'i’  -;.o 
■ 'i  i.'-i. ) “i'  ir  • 'J 


\ , ■ I 


II  - u 


n - 


Ub 

X*;, 


. .V  ..  o J Iwj  . i. 


f 


V •'-  V''  }f  XK-*  4-  4 ' 0 X ^ ^ ^ '^.4  . 


'’.  V » .7  j{.;riX''  X I H ( 0 vf«J 

, -V  q;’J-  rfXiw:.  y 'i 


,C  ,;;'0 

^ ',s;  •’,  N 


0J5  :-f;J  lo  O’Xv^'i' 


Y‘  A 


iji’.  ‘j;:.)-  .'■nxoi.i  "ii.  Jy;r>ri  .;;:sot^'^  rji,;xa 
.oovXOYO;.  cX  .*.4..,  <1  ,oX_  .A  difiX  J‘uc>'iXb 


11. 


path  ABOD  in  fig.  E.  Starting  at  point  A with  the  initial 
gases  at  a temperature  T the  combustion  is  allowed  to  take 
place  at  constant  temperature  T*  We  then  have 

As  a second  process  we  first  cool  the  initial  gases  down  to  ab- 
solute zero  and  then  allow  the  combustion  to  take  Jilace  at  abso- 
lute zero  so  that 

So  = 'Jo  - 

Finally  the  products  of  combustion  are  heated  back  up  to  T®. 

The  net  result  is  the  same  as  going  direct  from  A to  D.  Then 
we  may  write 


Let  the  instantaneous  specific  heats  per  mol  at  constant  volume 
be  as  follows; 

For  CO,  0^ , 

a,  -f-  2b,  T -f  3f,  t’' 

For  CO^^ 

* a >.  -t-  Sbj^T  t-  Sf^T’" 

Then  for  original  mixtures;  i.e.,  CO  with  just  sufficient 
oxygen 

|(a,  + Eb,T  tSf,  T*') 


For  final  mixture  after  complete  combustion;  i.e.,  for  1 mol  CO^ 


: ‘"a  - - Sc)  - - UJ 

Sv  = - S,)  - (Uo  - Uc)  (23) 


- U3  = |(a,T  + b, (24) 


^ ^ + 2b^T  tSfjT*” 


12 


Then 

- Uj.  = a^,T  + h^T’--^  f^T"^ 
Substituting  (£4)  and  (25)  in  (23) 


(25) 


dU 

=H  *H 


(|a  -a  )T  (|b  -t  )T  (|f  -f  )T  (E6) 


The  instantaneous  specific  heats  of  the  gases  are 


B.',  -t  2b,  T 1 3f , T a/  = a , + AR^ 

a^  2b^T  -^Sf^T**  a^  = a;i  v-AR' 

The  general  expression  for  the  entropy  of  a mol  of  gas  is 

s = s,  + 

= log  T -h  2bTf  ^ fT^  - R log  p (27) 

Taking  each  of  the  constituents  in  the  misture 
and  multiplying  the  weight  in  mols  by  the  expression  (27)  with 
appropriate  constants  and  adding,  the  expression  for  the  entropy 
of  the  mixture  is 

S r xs.  -b(l-x)s«  -1-(e-ix:)s-  -f-fs^, 

<-o^  cc  ^ 

+ [xa;  +(l-x)a;  f(  e-|x)a/  log  T 

+ 2T[xbj^-h  (l-x)b,+(e-|-x)b,-^fb,J 
+ 1 T^  £xf^-h  (l-x)f,  r(e-|x)f,  + 

-a[x  (i-x)  ( e-|x)  fj  log  P 

-R  j\  log  5^  f (l-x)  •log e->|x)  -log^ozi^ff-log  ^ 

In  getting  the  derivative  ^ it  is  convenient  to 
take  the  six  rows  as  separate  functions,  obtain  their  derivatives 
and  add. 


For  the  first  row  = s 

dx 


(CO, 


- s. 


''CO 


= -k 


For  the  second  row 


(a) 

fb) 


V 


X 


13. 


For  the  third  row  * 2(h^-  )T 


(c) 

(d) 


For  the  fourth  row  ^ ^ 

The  fifth  row  is  -R(m-|x)loS  ^ * -Sm^log  P 

Since  the  volume  of  the  mixture  remains  constant 
during  the  reaction  the  pressure  varies  with  x and  in  accord- 
ance with  the  following  relation  at  any  constant  temperature  T: 


P s 


m 


m' 


(27) 


where  P^  = pressure  of  mixture  at  temperature  T with  x 
Therefore,  log  P 


log  s t log  m 


and  the  fifth  now  becomes 

P 

-Rm  log  log  in' 

Tai:ing  out  the  term  log  m'  in  the  sixth  row,  that  row  becomes 
R j^m^log  m'  - X log  x - (l-x)  log  (1-x) -(e-|x)log(  e-Jx) 
tf  log  fj 

Adding  the  fifth  row  the  sum  of  the  two  is 

-R  j^x  log  xf-(l-x)  log  (l-x)f(e-l^)  log  (e-ix)^f  log  f 
+ (m-|x)  log 

The  derivatives  of  this  last  expression  are  as  follows; 


Terms 
X log  X 

(l-x)  log  (l-x) 

(e-|x)  log  (e-ix) 

(m-l^)  log  ^ 
m 

Grouped  in  a single  expression 
-R  log 


Derivatives 
log  X - 1 
-log  (l-x)  - 1 
-I-  log  (e-|x)  - ^ 
-i  log  log 


f 

'r 

I 


Z1 


(t) 

(:>) 


:;{  ,1^  ~ = 


J u 


S-  f 
- \ 


- /: 


,ls»  jw  C?  i Oy^v  ‘4.^  - 


r f ' ' ' ‘ ► »,■  J-  “V  > • ■ T~,  f ■ 

IV  I . —c  f '.  V - r...  ;ii 


i.: 

-X/ 


-',■.  ...Ow  ti'T-l .;  t 'X  .‘.  Ji.  wHu  .'.V.  C ; * ;.  or,A\- 

•„  ;.i  A ; liw  ' io  ejS*  nidtr./i. 

\^:'.  V ; 'I.  t OC-n. 


* • • *,..**  • ‘ » * r* 

4-'-W  v4..y  «»«i>  V 


*i 

it  ♦ ■«■ 


V ..1-  i 


C'.;  ■ .‘■“ 


' ,•!*■  iv  'io  3-auct'i  , « 

0.*?^'  “ tvX  =»  - 'SOI  , 

w -iii'-a  uo  -’X'xx  iWill:  uoil  let- 
lH  i;  r ;.ti  “ TX  'X  it- ■* 


, f1f{?C0..:  - eV.  .'  y , ->'X  I'w  1 

' » ^ ; ,C  ( ti'v  •"  J ’ ) ■*  ( -'“  - ^ ' ’ ' - ( X i - 


tsf 

o V ni  ic  ;:oI  iX':  v»  oii-J  'XyQ  ^rxXJ^-'. 


r t:X 


X : 


Cl 


r*  ’'1  . 

;v;I  I t , •■ 


» k 


wf  ...  lu  0 y '■  D'r  'r  q;;J‘  :-viijjX/. 

i'  -ol  1 (.-.  ; .nti'  (iC-  Ci  nol  (:f' C)  '•  >.  ’-'I  r. 

' ■--  K .L  (a  - ) • 

^ ■ ■ 

:..  ; . ii!^YX.ri3v XisJi  yffX 

C?  T ' ■ C /•i’-tiC  ■ Cfi.'x.5V 

L „ t!cl  '■'t  X 

J.  ••  \.. -X)  i'i'.  i*‘ 

...  vu>.  . '30 X 


(j:-X)  :^oX  (.''*•  X) 
30X  (x..'-  -'  ) 
••oX  {*’--) 


lit 


x:i.  i i :j  c u q;:  '•>  r X-^Tv.!  i a £ii  U © ' 1}'  ^ 


v.,-. 

I 


H'A 


X 


■'*—  --I— — , 

^ j x-i 


sol  f-'' 


14 


Collecting  all  the  derivatives,  multiplying  by  T,  and  adding, 
we  have  the  following; 

- -kr-(|a;  -a;)T  2 (|b,-  - |(|f,  - f^)T 

-HT  log  . IjV  i RT 


Prom 


- S ' + (|a/-  a;)  T +(|b,  - (|f,  -f,)T’ 

A-  T^- ^ , s^i-T  (Ja/-  a;)  - kj  - (fa/-  a;.)T  log^  T 

- (ife,-  fjT^  - RT  log  (29 


X Z'/n-jy  ^ 

l-Y.  \e-ix  ' r J 


C<7x 


'CO, 


At  equilibrium  A =■  0 and  — 


^ 

'i  P 


Substituting  these 


values  in  (28)  we  have 

R loggK  - ^ - (|a,'-  a')  log.T  - (^b,  - b^  )T-^(-|f, 

' (29) 

Where  G •=■  -l-R (fa/-  a^/)  - k. 

Substituting  the  expression  for  Ep  in  terms  of  x we  have 

R log,  7^  (-1^)^.^,  . (ia/-  a;)  log,T  - {^b,  - bjT 

-ft  ZlN-C  <30) 

Prom  (27) 


1 ^ ._L  xi  _ /_2SL  

P*-  ' 7^  > ~ TiT 

Substituting  this  in  (30)  we  have 


)^  (30  a.) 


»»  v' 


71  A 


H logeT^T-  - <5 a;-  a;>  log^  T - 

(4b,  - b,,)  T - fj.Zl’+O  (31) 

The  value  of  the  constant  of  integration  C can  be 
calculated  from  equation  (£9)  since  values  of  at  various 
temperatures  have  been  determined  experimentally  by  various  in- 


i ^ 


L.-  , -OVJ.  s : 


1*  > 

V -‘-  »'  1.  w 


Tv  1 


C^O  - T;'U:X 

r c i:  ■ -t 


L.ol  r;  v» 

'1  - ..  * 

*sr<“  *'  “ — ^*'" 


rac'i*'' 


f-  { r. 


■J.  , n(  X.  : ( ‘o  - 


Ip 


■') 


... } •:.  ( t. 

. ^ f 

ul  'a.-  -* 


1 5. 


..  ?Jb 

■ “ I.T  - 

■ - : . -':i^  - A 


'"P.  ^ 


-i- 


; u.T  ^ hi  ;/"•}  iX..dn; 


■( 


V 


: 7^,\ 
^-y','  V' 


jr 


*’  ••  ■*  • :’.•  ’ 


-.1 

r - 


* ' :.  ( \ 
) 


tl'^M  0 ® ^ UL'i'l  / ^ Xi.r*  r 
! , 

I y 4 c « V .-  » ) *. » X v *1  r ij 

),-7{  •\,’sox  (<  • - ■.')  -c  '• 


„4f  ■ 

. { c •'  -i-i  ) V Jlh  X 


M .X  , ~ 


V{ 


':t'  L 

,^^  r ..  ;s  . 

( ^*-  ) 

I. 

( .^'  ) . ri 


r:  : -..u-tu  L^- ...f  .*::  lirfuo 

r ..  V ' * ' 


W T 4.  i t • I 

(*'f:)  r.-'r'-L 


> I 


(CX  ) £i.l.ti-.ri^  1-iiXHrtUbU/c 


'.'■  ' ia  “ 

T • 


/ - 


■ ! 


::  -■'1{,--  - ■:;)  - 5,,  >,^f,-> 


'■3 

1"  I k.'.'H 


cc*  :*  ''u  r noi.0  A i ..i*  ,iO  •/ n/ v-'i;:nvo  v*-  uf/X  ;,y  * 


'■  . -4  .>>''  < 'iJX  _'  ' 

i<  ■ -.^Uv  . OC..I,.  {‘h'O  4-4  ■-".•r^IiLO 

’; .' i : iyp .;yo  //ofi-fijry.  - ti  jr**'  i v,."' 


' ' .(  , ' ■ 
'-pr 


'.-'ay 

- I ii;'-  i^.  -v  Xl 


15 


vestigators.  (See  Appendix  C.)  Also  the  specific  heat  con- 

anal  famperature  TJ. 

stants  and  the  initial  pressure  P^^are  known  so  that  for  any 
given  temperature  T the  progress  x of  the  reaction  can  be  cal- 
culated from  equation  (31). 


IV.  Derivation  of  3nergy  Equation  for  Case  I. 

In  the  problem  of  the  gas  engine,  however,  we  do 
not  know  the  temperature  T so  that  for  us  equation  (31)  has 
two  variables,  T and  x. 

A second  equation  in  T and  x is  necessary  to  de- 
termine the  values  of  these  two  variables  for  the  given  constant 

/ 

volum€»i‘gaseous  combustion 

J 

This  second  equation  is  derived  from  a considera- 
tion of  the  thermal  and  chemical  energies  of  the  initial  mixture 
and  the  mixture  of  gases  at  any  point  x during  the  progress  of 
the  reaction.  In  the  states  2 and 
p,  Pig.  3,  we  have  as  before  the  fol- 
lowing gas  mixtures. 


State  2. 

CO 

Ov 

N,, 


Mols 

1 

e 

f 

m 


State  p 

Cdo- 

CO 

0«- 

N, 


sum 


Mols 

X 

1-x 

®-|x 

f 

m-4x 


F,^.  S. 


At  the  state  2 the  total  energy  of  the  gas  is 

fj 

made  up  of  the  thermal  energy  and  the  chemical  energy.  As  the 
reaction  proceeds  to  the  point  p,  part  of  the  heat  of  combustion 
is  liberated  and  being  an  adiabatic  process  this  heat  of  combus- 
tion serves  to  raise  the  temperature  and  consequently  the 


* 


II 


. ' i '<  / 1. 

ro;. 


'.  .0  s'^^r  OJiv)  . L- if  0 r J .*  v’V 

•.I 

t^. .,  » i 1 vv^;.JW^- 

\ 

ritvi 


I.  t CC  U V.:-  '.l,l- 
4 . ...  :.u  m-v..';:  c'l.*"'.. 


?.'‘ t; 1 vA-j  - '*“£  'J.rw  /i  t.'J  . ,y  . T c.  w 


^ .1 


• ' 

- ; -^Px 

’ 1,’-  k << 

« ■ 

j 

.£<  ;•  .•  j L 

L .1 

- 

*«  • 

V 

J,  - 

' 's 

i i 

U-i.  7'. 

' * 1 ■ « 

» . .,  £«■'• -v  i , 

■/  .1' 

X . ; 


ij'i...  C'/i-  ‘VO'iui  Jtvj 

• A 2' 


, w ..'  ‘i .... 


■ V//-  ■ 


.1 


; ■ J X , 


, ■-'  O'  if  1.V  1' •>-.  JixiV  OLV  Slu.iLn”I  fn.* 

.’iv  u.?.C  v Ci/-.  i/’'^C:>;ruV 


- > i j':'-.  r,l  A.%iv.  t/.  i;  :.H,.u%ii  Liiix  ■ ‘ 

}j  x-.i.ii.  i.'.  L w ; u.v  i r.J.  •■  o ■ / .'v  &j  « adz  io,  n; 

.‘‘■'io-  v^-.'  ;:••  it-  .iv  •.v,.;'Vy .rr  CAJ 

.;/•.  .V  ■ s Miv  i,t  . Iiclj  P'.y’I  ri.Isj 


..i-C'  L' i -p'j  • '..i  i.'  . . ' '.'.'.-UJ 


i 


-Cj.  n^'.  ovp.'i  ©Mr  .'il'S  , q[ 


.»  f i 


-V* 

V 


‘X'  — r*f 


. Dwi  J 

C>^.- 

CL 


2ri  -L  . .1 

X 


oc 


X' 


f-  */: 


J 


Li. 


L 


P.Lx  iviut  ’i'p  t:CC:J  'h!w  - Jt:,  oiU  J'i 

cdi'  rrtr:'  iyi\z  io 

Iv  4j.Qii  b:\Z  V\y  nc::;o.:c'H 

Hll;  xC  •:fpO''L  ::r;p;-C7.:  :uL  ,j';L'‘j.-/i..  l>’  liL J.7.P0UiI  tii 

0/iJ  LXj*'A&£i^  L . lAk'' . iJ,  Xi'tiJi  (/' 4 .■J*.';5*r.O X j.'J*  yLx~'.  V <Tui.>. 


fiij-'  a 


rv7.i  j L;jU'.:vc  i 


16 


pressTJre  of  the  gases  within  the  cylinder  of  the  engine. 

The  energy  equation  (l)  holds  for  this  process; 


namely , dQ  a dU  + dW . 

Since  the  process  is  adiabatic 
dQ^O 

and  also  occurs  at  constant  volume 
dW  = 0 

so  that  our  energy  equation  becomes 
dU=0  (32) 

Let  us  assume  the  following  arbitrary  path.  First,  cool  the 
original  mixture  to  absolute  zero;  second,  let  the  reaction 
proceed  to  the  point  x at  absolute  zero;  third,  heat  the  gaseous 
mixture  at  x up  t o the  temperature  T at  point  p. 

From  (32),  then. 


- Ob)+  (Ug  - - Up)-0 

°8  - - {U^  - Ug) 

Ug  - U„  = xH, 


(33) 


Substituting  the  above  values  in  (33),  we  have 


Solving  for  x we  have 


- > 


rf.  - 


f ,-.i/  .... ij. 


Iv  t-Xi  .■  'i 


. .w  a ■-  - 


•.•■••  'I  f- “1 


J. 


‘ it  . .i . . ; ) i.  . 

. ; . j. ^ C>  .vA- 

% 


. i)  ■ i.  y...v‘  it'ji  *- 


i tin  ^ 
\ I 


0 it 


V >4  J. 

rh 


• 

o j 


I'J 


V/; 


t.'- 


Or 


k..  ..  J tj 


'J  kJ 


0:.i 


i • " U .i  n .i*  » i *,  i.  w 

f.  . 

>'  w 

n.iM  'tr 


\ M 


‘ . 0 ••  I 


J *1 


J » Si  V • 


, i.  »SO 

j)  ( ‘XJ 


) 


(V.  ) 


;}  - . 


'i  -iJ  ^ 


X.: 

n 


■ V.  - li 

- u 


. • ^ 


( (f. 


/ V 

\ 


i (.  ■•• 


. J ^ ♦ 


r.  ] 


( r lH 
G ! » 


( ... 


t,  .4  ‘ I ^ .. 


J.  cf  t,'.j 

iw-  ( 


- i r 


(, ; 


'■i-Vi.: 


I ^ V u.  - 


:j  - j ■ 

■*'  ' •' 

■;  - ,J  . . 

! ••  f.:) 

i - U- 

IJc  iJsL 


:■}  If-- 


/ ' - 


/ ■!, 


I ■:  <i  - ) .>v<:  • 

vr  >.  'i }.:J.rlo2- 


' I •.  ;■ 


17. 

_ _ m Tf  a,+b,  T-f-f,  t"*") -/r?  T^(  a ,+b,  -f , TD  I'ta\ 

""  • H TCa  b"*!?  f T )'-T|  a b ¥ f "T  ) ' 

Equations  (31)  and  (34)  determine  x and  T;  that  is,  the  condi- 
tions existing  at  the  equilibrium  point. 

We  derive  P as  follows  from  the  law  of  perfect 


gases: 

PV  - m'HT 
P,.V,  - m RT„ 


V 

P 


where  m'  m-ix  . 

V.  Numerical  Example,  Case  I. 

To  illustrate  the  preceding  points  the  following 
example  is  given: 

Assume  a gas  engine  running  on  an  original  mixture 

of  CO  with  the  theoretical  amount  of  air.  P,  =-14.7  lb  .per  sq. 

in.  absolute.  T^  * absolute  .Compression  pressure  =160 

lb.  per  sq.  in  absolute.  Find  the  maximum  pressure  temperature 

and  extent  of  combustion  in  the  adiabatic  Otto  cycle. 

Original  mixture  is 

Mols 

CO  a 1.0 

0».  e 0.5 

f 1.9 

m •=  3.4 

For  adiabatic  compression  we  have  equation  (14)  for  which  the 
constants  are  as  follows: 

= 3.4  X 4.51  =15.334 
Eb^  ^ 3.4  X 0.5666 -10^^  1.8890-10"'’ 


1 


r 


• . X 


•f 


V . 


V - ^ . U ^ - V 

SI  ■.  c -lx  W J.  .* . . X - *‘  V 


* ' •^r♦’. 


y.  c » t.' i -- A tj  >.  ’,‘J}  • \ i A/il,-.  \ -Ui-  t ' U'J  I. 

J , J : u t'i.'j  v*-J  lurl? 


y A 


. . J.  ':oaj 


n , ..-I.'fffrjxl..  *v 


-';;ir ’.lO*;.  uiivT  t.w.'ii.  ■ ,'i  i.  c.r.  „.i>i.IXi. 


m 


,I;‘.  i-  .ii  i-iv'  :..:v.i  ^:’S  o :. 

* , I*  T . ^ ^ 

xi.  # - »♦  ..  •■  *jJL' 


; iiovi  j « i ^ 

iM  ^ *(  *-•  ^ ‘ iy  it ^ ' V ' * 


0?.l  ii; ro*"-  •,iiix.JL(i.< ;'  (;.)h  * ./i 

■■ 


A-f  I . ' . ^ i f .’*»•»  '"’.  'i  • »\  ■*  ^’  * 

#c»  / .1  w j»  ^*'1  U J • V.*  .i  4.  l»  , k '»  *> 


■j.’jiiJ  0/W  i'.J'X'*’  00  Ij 
• i)v>  **  t !.  Q w.^t  ' • i'iX 
.MjJXocf.,'  r.'  'Cv>^  .ill 


. J!;.  (.r-  C-  wvJv'  :..  ■ J •: X'^.i  ;<•  t/w  »Ja  nv .f..t C 'u-;  v:g  3uC 


i.:  i.  o ' w « 7:1.13  1 Sill i 'i  0 


-■IdIJ 

ft  - " 

<1  r 


'i: 


QO 
- 0 


t; 


d 

ii 


no *i,o  ; (-^T)  ••■.i,n  & ^.v  . ::. 

or,  iroXt..i/C-,'XC7-;oo 

oXij 

10% 

,‘  Utt‘ 

t-'-d 

! 

T 

I't''  .Ji  Xj.-  X 

0f\ 

i 

• 

l * 

(.1  00‘e-c.x 

(;I‘  dcdiil . C r 

0 


18. 

AH  ^ 1.986 
mAR  ^ 6.749 

Substituting  the  values  for  T,P,an(i  Px.  equation  (14)  reduces  to 
§.5699^Tx  =^3.14536 

Brackets  indicate  logarithm  of  the  coefficient. 

By  trial  = 1£50°  . 

The  constants  for  equation  (34)  iriiioh  is  the  energy  equation  are 
as  follows: 

a^*4.51  a^-5.42  =121. 250 

b,  = 0.2778'10'‘^  b^=  1.75*10'^  m -3.4 

f , = 0 -0.16 'lO'^  T^-  1225 

Using  these  constants  equation  (34)  reduces  to 

3.4T(4.51  +0.2778-10‘'^  T)-20.643 

l2T72gtf  + Tri. 345-1. 333i  'lo’  T +'0.16 -W**  5>-) 

By  substitution  the  following  pairs  of  values  are  obtained  from 

equation  (A): 

T 5200  5300  5400  5500  5600 

X 0.738  0.759  0.781  0.803  0.824 

Equation  (31)  reduces  t o the  following  with  the  substitution  of 
the  proper  constants;  (See  Appendix  0.) 

log  7^  ( = -6.5661  log^„T+1.335-lo\ 

-0.08*lo'V-13.1+iJlog^  tlog  tJ  (B) 
To  obtain  pairs  of  values  from  equation  (B),  assume  a value  of 
T which  then  definitely  determines  the  value  of  the  right  hand 
member  after  which  solution  for  x is  accomplished  by  trial. 

We  have  the  following  pairs  of  values  from  eq.(B): 

T 5100  5200  5300  5400  5500 

X 0.825  0.802  0.777  0.750  0.723 


I 


( vO 


t .'S 


‘ V..  f V • w , *.1 


V 


S7 


• oX 


j . i.  t. 


ii. 

.'u.  I - ,, 


J.f! 


X ‘ X < 


-i  W -i»* 


'J  O ^*4  V-' 


: c •VOxi.!.'- 

. '■  X ' 


.(J 


0 .-  .1 


.-TOc  ?c.oiIu  >u*ir;iwf 


. t: 


I A « ..  U 


; ► rrf 


<•' 


ULi^r.  C .'.'j  : 


: ' > i ' V J.V  -•  c 

•r  '■;  1' 


'f.  J...  . : 


/ - 


0 •liiA!', -.a  : 


; -.I  I:-..  . 


V ■i'.viX 


-. ..  i\  L.  r 


: cl| ; X.  . 


'.  i.  • ■ >.  . ' 


.1.  ' V V , 


.)*  V r V 


' u",  V'  <,t  ’ < (.?  ’ . 

.’r.v  .t;<  i ;.i  -'■)  iff:--...  J:.  L 


.<  V 


1 i ‘ . « 


J.  J ,/i  J. 


X- 


it 


v'  V - 


. J 


11^ 


It 


1 


( 


19. 


Plotting  the  values  for  T and  x from  equations  (A)  and  (B)  as 
shown  in  Pig.  4,  page  ISa,  the  intersection  gives 
T - 5335  X - 0.767 

Prom  equation  (30  a.)  we  get 

P = 

m ■ 

Substituting  values 

P - ^ , 1^0' 5 ^3  S 

3.V  ' 

P = 568.1  lb.  per  square  inch. 


VI.  Derivation  of  Equilibrium  Equations  for  Constant  Volume 
Adiabatic  Combustion  Involving  the  Heactions  CO-^^^-j-CO^ 
and  C0^-^C0+-^0^  Case  II. 

In  the  case  of  the  actual  gas  engine  running  on 
carbon  monoxide  and  air  we  have  the  additional  reaction 
C0j^->  CO^-i-Oa.  from  the  fact  that  some  of  the  exhaust  gases  are 
left  in  the  clearance  space. 

As  the  combustion  proceeds,  CO,,  is  formed  by  the 
burning  of  the  CO, and  some  of  the  CO^  in  the  initial  mixture 
is  dissociated.  We  shall  let  x , denote  the  progress  of  the 
CO  burning  and  (l-x^)  be  the  progress  of  the  CO,  dissociation. 


Initial  Mixture 
Mo  Is 

CO  a 

0 e 
f 

m 


Intermediate  Mixture 
Mols 

C0>-  ax,-f  gx^. 

CO  a(l-x, ) + gd-x,.) 

Oz.  e-i’ax,  f-i-gd-x,.)  = e 

N,.  f 

m-'i^,  -^-t^(l-x^) 


Let  E/  be  the  reaction  energy  of  the  CO  reaction. 


tt  n 


COjj^  dissociation. 


-1 


^ 1.  . - - * 


■|  V ; -v: 


..vi^ 


» J'i  Hi 


' a 

I ’J 


•«  f 

00  ) O Lv'. 

- nv  ^ 


V^A 


u 


• \% 
i \ 


' L-:v  '-f  J.J  L-y  iOj^ui/O 
s'  it 

-^  • - V u 


V/- , 


.0 


a- 


: ; • . U '/  b fl*.'  0 C.  ' C'  V J w J1  _ 


■'©'  01*  '.I 


.'t  ■ * ■ . ■*!  C i.'.' .■  0 "i*  .'  V ] J._ /] 'v.v.-  . 

j i:t) 7,,. , •'/,.  V, i -O.^ 


wo  -^n 


• »w  : ^ .1 .. 


■ J Pi'  -.  *:::  L o.v»iJ  c O 

•T'  »..''  iULD  , 


- 


‘'  . CO  l/ful 

*C  T ;;  ;;  t 

i V 

Oj-  cTDi.;  ..',v-J'  aroi^  « • ^uo 


A<4f  *V 


• 

i.i 

''-f.(..;'r.' 

iit^J  X’l 

'•••-•c  " =i  , 

•.-t.V  i 

. .>  , 

i.;  .!  ..'V 

1 

V 

' liioir;- 

j*i;J  ni 

,..0  : 

;•...  'IC 

«nO 

'*  • 

ndS 

'^0  ; : ■ 

X .’XUd 

c,-.e1>0T 

0'‘4‘ 

. 

:i  ?<  " 

r 

■■- 

*£>0*ifi 

ii-  oi 

-•>-  -^-J  ^ U V- 

■■  , ■•• 

- s r*^f " 

4 

'<■*1  ■ V ; 

^ M ur 

■ ( 

n 

'm  ji,  i >''t 

^ ^ ..  i 

r .*■  *• 

w*.  k 

i 

■'.  .ki 

.•  ^ 

VJi  JjblXisil 

' • 

!*• 

‘ ..  J,  '’ 

{.--I)'  ■ 

. , * 

- * i;  \j- 

iJ 

00 

' ^-X;a 

r 

fj 

-»0 

( ,j.  - £ )7;i 

X 

f 

= w 

> 

4. 

n- 

^00 

^ ^ 

~^«3 

•H  ^ 

♦ a 

i);50'’X  OO 

A -f 

•w  . ■ 1,* 

00  05^- 

; i' r ' , 1 

XI  c 

I »ifx 

ad 

0 

x%i' Xh  .00 

»» 

n 

, *» 

i> 

H 

“ n 

1 \ 

» 


( 


Fig  4 Case  1 


r"- 


— f - 


page  fSa\ 


0^ 


1 


§ 


CO 


.Ll.Lt._l. 


CO 

!5 


s 


'X  $enj&/\ 
t: 


te 


J5 

c> 


SXF.  'Z! 





-a 


*'•  p 

• ■> 

N. 

1.  _ . 

c?. 

,.f;; 


rcf 


\Si!  . ’i,Ti  ,;i> 


r^. 

<• 

b 


t "il'  X‘ 

V? 

it 

i^.'j 

s 

£>  ' 

b-  - -■' 

* ''V 

!V'i 


20 


Let  S be  the  reaction  energy  of  the  combined  reactions. 
Then  B -B , + 

Prom  equation  (15^ 


y, 

aA,  dx, 


ViThere  A,  is  the  driving  force  of  the  CO  reaction  per  mol. 
Similar ily 


E 


X 


/'  Xj, 

gAj,dx 


a 


where  A^  is  the  driving  force  of  the  CO  dissociation  per  mol. 
Then  we  have 


E = E,•^E^»  J -tj  gA^dXi 

Taking  partial  derivatives  of  (35)  we  have 


(35) 


if  = aA, 

(36) 

(37) 

Since  at  equilibrium  the  driving  forces  A , and  A^ 
are  each  equal  to  zero,  we  have  two  distinct  equilibrium  rela- 
tions established. 

Prom  equation  (19)  we  have 

^ ^ _ T-^-^  - ^ ^ 

^ ~ 3 a,  3 X, 

Since  energy  and  entropy  are  additive  quantities, 
the  expressions  for  the  energy  and  entropy  of  each  constituent 
of  the  intermediate  gas  mixture  can  be  obtained  separately  and 
the  desired  partial  derivatives  taken  of  each  expression.  Like 
partial  derivatives  of  all  the  expressions  for  the  separate 


(38) 

(39) 


V 


21. 


constituents  of  the  gas  mixture  Vvlien  added  together  Vfill  give 
the  partial  derivatives  of  equations  (38)  and  (39). 

Sco,  = a ^ log  T -#•  2b  f^T*' j 


- r[^  , -f-  gxjlog 
Let  s a/  log  T 2b^  T f-Jf,  T*" 
log  T 2b^T 
From  equation  (27) 


/77' 


J:  p 


(40) 


m' 


ra 


Substituting  these  valies  in  (40)  we  have 

- a|ax,+  gx^Jlog 

J ■ -°s 


ox,i-3^i. 


fir 


P - aR 


ax, 

l^*"  = log  - gH 


For  CO 


= [a(  1-x , ) ■^  gf  1-x^  )J ^s,^^  + f,J 

- s[a(i-x,)  t g(l-x^)J  log  p 

Hr  - ^ J ^ 

“ -s[s.^.  + + gR  log  -■— p+gj 


^^<5. 

a Ax 


aR 


For  0, 


3^,  = [e-iax,t  Jg(l-Xi)j[s.,^  i-  <f>,J 

-H[e-l,aK,<-te(l-Xi)J  log  g..-i°*- ^i3C--x. ; ^ 


nr 


H:  = -H 

H:  = 

For  R 

b 

_ 0 
3 /, 

^ -^Vx 

3Xx 

tT) 


/ -^'  ■(’  'si 


•ga 


- 0 


The  energy  equation  for  tho  total  intermediate  mixture  is 


Vi, .v  :;VC'  c-*,  Ln  e^'T;  w';:  :•  Ut-tv  {, 

•\-‘'  ^ (-V. ) -tr.T.  Zc  . > *io-- - "j  Vi ^ ■-ni.fio  Lcl^ia>.  r ;’ 


'.,i-  i ' c 


- Ov-  X wl  . -r  ? ?:::  * 

~ ^ ,£  .q,  .>  ^ . 


• * ' ' ■-  (id  'S  4,  '^.ij;; 


^ ,ua  -•  ::  ^ci'  is 


> 


<y 


‘-r^  e>*  { ■ i ,.:  „oi  ■:  'viuJ'  ' * ...sfii JUiibi;.. 


cX 


r-  J i 1 , ' -'i 


cl  .-' 


'■'.V  jtil 

1*  - ii  ! 1 3? 

'"1.^  _xT 


OC  lo'i 


( 


fit 


i l;^/i.,..lL 

{fi 


V (,  .“nnj 

;,iC>X  .,k'  V ' 

«r  C 1- 


£ .3^-1  ) 


».  / -j  ^ - 


V 


■ ftr* 


..’vTj- 

r ^ 

w *10^ 


. -A 


;iX:i  =-i)..  ,■  r 


■■-■  tf  »i 


- ,tr:,  C-'  -' 

‘V  ■ , 


^ *.  ,4.  r ^ , 


«.  ra  • rv  • ^ 

•'  I ■••" 

'L 

• 4 


.4 


al 

* I ' 

4 


’irMlI- 

■•T?. 


r. 


\ J\  r 


H lola 


0 


aJ  j^xM;r- iTj-fti  L^Co. 


.-.,«  '•  . 


CiX  i'.c  Si  erfT 


22. 

-h  j^e-i-ax,  t ig( f 

" ^ ^ "2  ^<^3,~~  ^Co^~J  “ ^ 

~ S -t  ^Cd)^J  - 3 ^V 

Collecting  the  various  results  we  have 

If,-  ^ RU,  T-R] 

-1-r[-|«  + <^,  + /?/^3  ^t^±s0i2^r  +7?J 

+ T [-i  S»^-i  i+in/og^.P+iR]  C^3a) 


a[R'3x,~  §x,J ~ -(R-i^)T-+(‘fxi‘l>OT-RTlf 

J-fr  ^ j ^ 1 - sam<s  ex  pres  sion . 

3 1 T7,  _/ 


Prom  ( 39  a)  and  ( 40  a) 

J.rr^^'3C^l-_U  Ft 1 

3^,  Tx*  J 3L 

or  1~  ^ & L 3 B 

« 5x , ^ ^ ^ ^2. 


Prom  (36)  and  (37) 

Combining  these  with  equation  (41)  we  have 


^ *'3- 

If  the  driving  forces  of  two  reactions  are  equal, 
then  under  the  same  conditions  the  two  reactions  will  establish 
the  same  equilibrium. 


wr*: 


*r-v\*j-^',M>l  „;u*- 0 ' .t* 
'V.  .,«!  V t'CaJft'WoJ  m 


i^^  s^  ■= 


^li  . 

' L '-'"J''  ,xs, 


‘jvr.if  uiwitay  feJl  ‘5ni^o#lj:«0 


♦ 1 

is  - «d  i. 

.K' 


y 1,-  •r'-^ 


,-■*!  f( 


.f’ 


■ * y V?'^  - y 

k « 

’ * ' ' 

r c-t£j  V <;  9*.  ] I 


‘4: . 


<V''» 


) 


A>  ??i  ,<,r 

7k.  e-^  5 


, . — «—  « . wiiS?,^''.  •«“  J- . 


■‘  " ' ♦» 

' ...  .fc  ■ 

.t  , * ' . 

1^  ■ f,v^ 

iy’. 


vji 


(»  Cl)  (a  ^S)  mo«i*f  ^a 


«*Vi 


4 A %• 


-*  -'r-.'.,t 

•m 


ur.uj 


n? 


* “ ‘*  * M'l  ■ ''  ' «>  r|  • • i!’  ■'  •» 

iv^)  bsm  im) 


-r:  e.  ■ ■'  A'*'  ^ 


'V  - ^ .x&  . ; ^ 

o?sx(  0m  iX>)  d’AJt^i  00061!  ignffl tiitei0O 

' *yf  ' ^ 

M yl‘ '0^^S^A  Jtk  .Ui,>il‘'Xf.  m A 

1^  ; 


,l0Myc  i»au  sitoij'uja^'t  0*1^  to  tto'ioi"'?«»iTiyii  nib  ii 


XXjtw  «n^i /& I ex  ov^j  ■ e in>J;#t'ii^ip  ,«E(tf  iforf^ 


£*w 


•tnifi  iKfi  !lap0  gtii»4  9£f^ 


23 


Therefore  at  equilihriuin 


X - X,  ^X 


Prom  equation  (39)  at  equilibrium  we  have 

0=H,-(h-iR) i*.) T-RT lojg 

Co  X p 

/TV  ' ' 


/-X  [g'fJ 


(a  -f-s)  /-X  -p/  e' 


'R  ,p- 

'CO  'Or 


= /T, 


• '•  F(  /05,  /Tp  = f?  - C/r - i i <^0 

Substituting  the  expressions  for  (^^and 

+ r+  zi,T+^f^T-  J a,%r-3Ar-^  )jr' 
-(A -3^) 

Collecting  terms 

H,  - ^ -(ia;-c:)/c>3T-{ii>.-b,)  T-i(i  tor 

-\k -in -(fa, -02 

The  above  equation  is  identical  with  equation 
(29)*  The  equilibrium  relation  between  x and  T is  therefore 


^ / -3tC«  ^ 


/r? 


±3  ^x(F-^3)^ 


mt 


= - x(orO/<=>9r-^i,-OT~i(ifrn)r^  c i^Z) 


1 fV^  . 


“ ► ^ V,  * ^ V ^ ^ **  ^ Q ' 0 w t oii  T ■] 


eT*«i  ew  mXiu-tllifpo  it^JlSt)  aoi<yojj^0  mpvi*l 


-^(♦-f  iKi.- A) '•VA-t.^ 


<ar  ■’&,<; S.X^, 


sT. 


^K4')‘a 


U 


i - a")-'  ^ijcA  f\  ’ ,"  ii,^j4 


xoi  aii'^jl*(eoTi^.4j  ©fL**  jp.. 


?,  f T:jt  1.  +.:t  ^>;  .t^v^S  > i-.  , . 


tX*’ 


. A > 


• 1'*!..  ■■•  ‘ .1 

• -‘  .1;' 

•*'% 


fl 

lluW  ■ ... 

, ’V  •.  ■ . ■'>!•'.*  mmml  y-  ^ 


mUi'^9  A^l'»*.l^ulXn>jpX  iK^iXaojpt  eYorfa  *.{t  _ „ 

dXOlbiOilJr  Bi  r JbniB  3c  cr^Bv-XatT  aot^.:Xe%.  fSuixdZXXii^fl  ©nT  -.(SSl 


{^ C**'' -«-..■■■  lltilM^^  lt  |l  ■ •"''  ,. 

\t;^) 

^ \ ' ' i^'"  ”''' f ■■  i''^ 

(.VV)-  ^ 


; 

M . ■ 


Vs5v*  ■ 

■lIK  ' - ' r7»  i‘Nt..  > ,!» * , 

®^3liitf  a’S  . '"  V-  ^ , J > 


' <i'  '%■-*»'  ®*  •- J '^..  . 

■j?  , , .A  , ■'“.%■  -V.  ., 

J-  II  *'  Jj  'y:>j  ./:.. 


1 .wt' 


VII.  Derivation  of  Energy  Equation 


Case  II 


24. 

The  energy  relation  between  x and  T is 

oxtic,-^g  0-^)  /ia=  (a+£)y.T(a:,-h  bjTif^T") 

[nrfig-iCa-fgJ  yJT[a,+b.r*f.  T'J-u^ 

U^.(a4eif)  Uo.^hlHV)  + {a,^-  4 T,  f ^ T^) 

•><  [(‘^ -^•3)He  ~(p-t3)T(a^,->  bj-i  fj^)  + l(cn-3)T{a,-t 

= g Ha- Uj,  i-(m  ->i:g)T(g,i-b,T-tf,  T'^) 

(a-tg)  -(a^-i-b^T+fj,r)  -t^(a,  + b,T+f,T’-)J 

The  solution  of  equations  (42)  and  (43)  gives  x 
and  T for  the  combustion  of  the  gaseous  mixture  containing  CO, 
COj,  , Qj_  and 

VIII.  Numerical  Example  Case  II. 

The  following  example  illustrates  the  procedure: 

A gas  engine  runs  on  CO  using  the  theoretical 
amount  of  air.  P,*14.7  lb.  per  square  inch  absolute. 

T^  = 650°P. absolute.  - 150  lb.  per  square  inch(«^5>. Assume 

that  10^  of  the  products  of  combustion  are  left  in  the  clearance 
space.  Find  the  maximum  pressure,  temperature,  and  extent  of 
combustion  in  the  adiabatic  Otto  cycle. 

The  original  mixture  in  the  cylinder  is  as  follows: 

Mols 

CO  1 . 00  a 

0>  0.50  e 

N,  2.09  f 

CO,  0.10  g 

3. =•  m 


T 


*■  ■ 1^1'  : X ■ w ,. I njiv.*  .C,*  ti'-'v't  i.- 


/ Cr:^a  V „tA  y'n 


' . V f 

V J.  ^ .1 


J^.v)  i ;■,^^  'q)- 

* ■ I ■ ' ■ ■ ^ S'.-'.-  ^ ■ -,•  ! 

,('T  :'\  A t i "T^)  t i ^ ^ ^ 


fai  f i'T'  9 : 


.'-  \ 


iJiilce  ©rtT 


->j  • 

i ,■^>  'rf.  li.iv -X/u  -...'tJ  5.,J  'XCa  a 


0 ,^j0 


;.;  ’ ■•■■-'  V.' 

. “tn  ^ J.,U/Ctfie 


• S>lJ  »/  ..  irii,  ijQ  • i Oo0  - • v^'*' 

' 1 . :^■^V  • 


r .,  Cl  ^ .'-v-ivUi  .„  cv  'i.,/ Cw...v*‘iq  t,^.a,^o.%X  Xu.*"fJ 
cj  .1-^  ,,  ..^  . ..jisrxiiil.  aiiX  Xiil'i  . jo-ite 


, V U\»  V V(  ^ j.  ■-  . 

UilJ* 

'TCiXuuClV.OO 

V - * 

, w {,*•£'. o 

(it 

„':£xc:  Xc 

iU:5i'iO  v.i'X 

■ -:XoM' 

' * s’/' 

C. 

OC..  £ 

OD 

' 

0 

\c  ^ 

• 

\ 

1 -.‘  •■* 

’*  • 

-.OC'  ,••  ‘ \ 

* 

. ».  , t 1 

!•' 

' VIT  ■■  '«  SJ 

25 


The  constants  for  the  adiabatic  compression  ectnation  are 
= 3.59  ' 4.51  tO.lO  • 5.42  - 16.633 

2b^  = 3.59  • 0.5556  *10'’-^  0.10  •3.5*10’^^  2.345 'lo'^ 

=-0.024  10 

m4H  - 3.69  - 1.985  = 7.325 

+ mAJR  • =23.958 

Substituting  these  constants  and  the  values  for  P,  , T , and 
in  equation  (14)  we  have:  (Brackets  indicate  logarithm  of  the 

coefficient.) 


log.^T^  + [5. 6284^ T^-  ^0.6385^t2  = 3.14879 
By  trial  T^  =»  1250®P.  absolute. 

Substituting  the  numerical  values  for  the  constants  in  equation 
(43)  we  have 


-JO^Q^  ^ 3.7^  S'!  O.Z778  /O'^T) 

UO  ^ ^ -/.3333  /<y~^T  -f-o./6  • 


(c) 


By  substitution  we  get  from  equation  (c) 

T 5100  5200  5300  5400 

X 0.810  0.834  0.855  0.877 

Substituting  numerical  constants  in  equation  (42)  we  have 


By  substitution  from  equation  (B) 


y log  Tj 


T 5100  5300  6300  5400  6500 


X 0.826  0.802  0.778  0.751  0.724 


!T- 


. J i!>.  i 


..V  ••  .:,c 

% 


ci-  .. 

Ll 


I 


A > , I - 


-4.  \«*  V 


'.-I  *' 


_0  TL  . r'C  'J.  5' u J ^ y '.  J. 

)';'  ■ 


rroi.' j-i* a:  J i.)'.  / / i#  . j.v  »[*'?  Uedc. 


■ ■ ,.  • •;  : - 0 . (&'». 

• » ‘ • i "■'•♦v'  , 

•. ( T \ S .«  ■ ■ • 2 

^ V ^ A1 

V <,.N  V 51  "2-.*^  • I r^,.VA  ' /* 

v' 


\ ' \ >V 


( .;•  i j ' j J .iit.o  >fi>  - i.'t  nc^'.’  u i^acfira  fS 

. « 


I I 


c "€^  Ooa^  ■ CK  la 
‘.0  ci,-^o 


T 

z 


. ijv  ;;i  I . • { :■. i ) r.clw  4J*  u *.l  . ;rv-.-f:A»^o  I;«ol"ref: 


fr 

V 


•'’•  .,'.>■  ■i.  •■  ■ ■ ..  . ■ 

r."*'  -t'  .-  V.  >-■.-  ‘ 


•V, '■-'>'  ^ ^ 


. ■ .ii' 


(ij)  ;;'  ,cvi»i.’pt>  r.  ct,’''  ['i:' \:£ 


CrOW  CCc'.  001'^ 

I..V.C  5:OE.O  or..‘.o 


T'. 

$■ 


1,;'^  : ■/ 


• I . 


i 


27. 


The  intersection  of  the  graphs  of  equations  (C) 
and  (D),  fig.5,  page  26,  gives 
T = 5135  X 0.819 


' m ‘ TZ 


71 

•p  - - -k(l’0  i)  0,8 13  /5~0  -^/35~ 

3.6  7 * izs~o 

P * 549.4  lb.  per  sq.  in.  absolute. 

Equations  similar  to  those  for  the  reaction 
CO +-|-0^ :^C0^  can  be  derived  for  the  reaction  -l-O^;?. H^O  by 
the  same  method. 

IX.  Derivation  of  Equilibrium  Equations  for  Constant  Volume 
Adiabatic  Combustion  Involving  the  Burning  of  CO,  and 


CH^  and  the  dissociation  of  CO.  and  H,0 


Case  III. 


Y/e  have  available  experimental  data  on  the  equil- 
ibrium constant  for  the  following  reactions  only: 

CO  ^0^  ^ CO^ 


-10 


CH  20. 


H,0 


CO^T-  2H,0 


Y/e  are  therefore  theoretically  limited  in  our  discussion  to  such 
gaseous  mixtures  as  involve  only  these  reactions. 

The  most  general  gas  mixture  that  involves  only 
the  above  reactions  is  as  follows: 


Constituent 

CO 

CH^ 

0,. 

CO^ 

H;,0 


sum 


Mo  Is 
a 
b 
0 
e 
f 

g 

_h 

m 


E8. 

In  the  comhustion  of  the  methane  constituent  of 
the  above  gas  00;^  ejid  H^O  will  be  the  products.  St  the  high 
temperature  produced  some  of  this  and  also  the  H-^0  will 
dissociate  so  that  in  reality  the  products  of  combustion  of  CH^ 
are  CO^  , CO,  and  0^. 

V/e  shall  use  the  follovang  variables  to  denote 
the  progress  of  the  various  reactions; 


(1) 

X,  progress 

of 

CO  reaction 

(E) 

y/ 

TT 

tr 

(3) 

z 

n 

n 

CH^  " 

(4) 

(1-x^) 

Tl 

T! 

CO^  dissociation 

(5) 

TI 

tl 

H^O 

(6) 

( l-x,) 

u 

n 

CO;^ 

) Produced  by 
)CH^  combus- 

(7) 

tt 

tl 

H^O 

) tion 

In  determining  the  intermediate  mixture  consider 
first  the  CH^  combustion. 

(o)CH^  f-  ( So)0j^-»  ( Ecz)Hj_0  -^-fczjCOj^  ■+•  c(l-z)CHy 
( Ecz)H*0  ^ Ecz[yj  Hj.0  -^^(l-ys  )0^  (l-yj)H^J 
foz)C0^-^  czjxjCO^-f-  i*(l-2^0,.  -h  (l-x^)CO^ 

We  have  from  the  various  processes  of  the  combustion  the  follow 
ing: 


o 

o 

o 

CO^  H 

CHy 

1st 

reaction 

mmmmwm 

a(l-x.)  -iax, 

ax  / - — 

End 

Tl 

b(l-y,  ) 

-|by, 

— by, 

— 

3rd 

IT 

— 

“Ecz 

c(l-z) 

4th 

disso. 

--- 

efl-xj  2g(l-xJ 

gx  ^ 

— 

5 th 

IT 

hd-y^) 

ihd-yj 

by  ^ 

— 

6th 

TT 

— _ — 

cz(l-x,)  i-ozd-xs) 

CZXj 

— 

7 th 

IT 

Eczd-y^) 

oz(l-y,) 

ZqzJj 

--- 

u 

i I-  •: 

,0  ^ i 


•»  . , r ♦ , ri-  . 


uXIOJ.  -Jj 


w - W S 


4 %J 


'-  - .'7‘-*  t X i.hf‘  i.‘^  vh  i.T 

'v'  uv'  ilil-.  v-V.H  lr.r.  ^'.0  t>i('  «>yc f'.^. 

-Xc;  pli.; . 

. e*r  .‘X  P/m.  ot?  igaiuf-sr? 

• Li  . , . . L . ■ 


■N 


i . 


. . .1  Jt.  '13010  *.  J 

J . .ri(j  ,.)w  (X) 


>•  ‘ 
f , 


iioiJ  ^1' 


' «'  •.  x>  ^ ^ 

^ •»  ..  f 


.1 1 f ^ A. 


i>  w « i 


/ i. 

V 


''  X (o) 

" ( (M 

!•  .-X).  (:.) 

'•  ( -.:)  (a) 

(.-X)  ..•) 

nl  ■, 

' A 1/  v”<  {.T!*!*C 


y ‘ ^ X ) L ■'■  V.  , / ^ 'J^  i "t>S  ) 'V  " ,'0  { f;^  ) r ( 0 ) 

: .- - ) • \"l)'  ’ ^-  ,-i  LL'i  ' 

; • V -'Ll  ~ P’  “J.  J.  » , ' ^ 


.J. 


» . 


• W-' 


^ • H'  i»  » ' ' 

■ -0  %}Ji 

'X  L / 4). . . 

klq-x'z  a>vjx 

i o-i 

■ -,yi 

'■  y ‘ 

';nl 

u 

i ■'i 

> I 

'iff. 

' 1 

'' 

'f 

f , -5  ~ li  /'  ' 

i f 0 U.  u -<Jf  '• 

PtX 

' - 

: 

4 -X  ) ::; 

?» 

i.-i:a 

- no, - 

-;  - ~ 

f» 

xlCl 

1.  v.-j: 

{ - c ; , 

{ 

* V X W 

M 

- ’)-o: 

( -D-u 

.rt  ^ - 

n 

.u1  b 

V X.  i !'  0 

f 

- - V t, 

y- 

u 

■•I).  0--1 

If 

iurv 

-•  no 


29. 

Collecting  terras  we  have  for  oiir  intermediate  mixture: 

00;^  ax,  -t-  gx^  + czx^ 

H;j^0  by,  4-  hy;^  2c  zy^ 

CH^  c(l-z) 

CO  a(l-x,  )+g(l-x^  )f  cz{l-X3) 

b(l-y)  -/-h(l-y,^)+2cz(l-y3  ) 

0;^  e-*-i^(l-x,  )+ib(l-y,  )-hi-g(l-x^)+-|-h(l-ya)-f4-oz(l-Xj) 

+cz(l-yy  J -2cz--i-(  a+b)  = e' 

^ 

sum  = m 


Since  the  reaction  energy  of  the  total  is  the  sum  of  the  reac- 
tion energies  of  the  reactions  of  the  various  constituents , we 
have 

dS  = A,dx,  + Aj^dx^  +A^^^-h  A^dy,  -h  A^-y^  -^A^dy,  -hA^dz  (44) 

Then 


9 E , 

9 X,  ^ 1 

A 

3 E , 

II 

9 E , 

0 E 

9e 

9 Z ~ 

(45) 

Also,  as  before 

= TdS  - dU 


then 

Let 


<3E  3U 


= a,^log  T ■+2b,  T -t-jf,  T 
^4=  a^log  T +2b^T  +|f^T 
s a',  log  T -tShsT  ^-IfjT 
<j>y=a;iog  T +2b^T  + |f^T 
- aj-log  T + 2bj-T  -f  -^fj-T 


i.e  • 


i.e  • 


i.e. 


i.e. 


i.e. 


for  and  CO. 

for  CO,^^. 
for  HjO. 
for  CH^. 
for  H^. 


n 


S!t9fJ0Xi  ktioo  j^OO  ^ 


■A 


♦*ir 


i 


-i 


r 


:1 


/??■ 


i1 


?>r  [ 

5^S3-  [ 


JAito/ii^’WiOO  Oj)i  &d4  ict  4 


iJV 


H4-  \ 
H A 

5^ios-[ 

K.V5S-[ 


* • . _ $ 1 

**  1 


^ i 11 


S ,'.J 


OltOO  Wf^  -XO^  jr 

r - F '‘j  I 


^ ^ ” v*=^  '1  ' 


•*,1  'V 

1?  ■ 


. ■.  - '- 


,/tx;  .»,  i 


ra  . - I I 


anuarx: 


31. 


For  the  CO  constituent 

s„-  ^6-x.>^g(l-x.)^ C2(hx,)]ls.,.i  4,] 
-R[ 


J log  pj 


Ifr*  -9  l 

-C2 

I 

lj‘°= 


3 Sco 
<3  X, 


4,]  4 aTi  log  [ 

" ] -ioR 

[ ■ 

J /og  [ 

" ] 

L ■ 

] iCzR  log  [ 

•■  J -tczR 

L - 

]-c(/-xjl^/og[ 

••  ]-c  (/->,>? 

Por  tne  constituent, 

^ CZO -(/,)]  [ 5,  4 J 

-T\[  J -n,)pj 

■■  J 

-/?  [ " J 4 /i  R log  [ - ] 4 /?  7f 

••  ] +ZCZ~R/ogl  ..  J 4^czK, 


<3  5. 


;m.  = 


- -^CZ  [ 


For  the  0^  constituent 

6f-|a(l-x,)+ih(l-y,  )+igfl-x  Jf^hd-y,  )/.ioz(l-x,) 
■f  c z ( 1 -y^  ) -2c z -i-(  af b ) 


«./  hi'. 


J . . J I o -C.  . . ^ 


* 0=2* 


t 


A ■ 


' »►  ’>'-  ♦ 


j \-\ji  n ' ■ '^_ 

) ^c.VrVs,3-i-  ^ \ 

/ 


l 

f 

' ! '■ 

><; 

.vc: 

I 

^ ]>-. 

X - fc-  ^ ^ 

? ^ J . 

•* 

^ /; 

■ ,r 

» ]<- 

j r ‘ 

•44 

t , 

] '’'X- 

r 

■' 

1 

■ -S,-!-  i 

•.4 

4»^-4 

' V 1 

.-».'> 

• jr 

. .1 

\ ^ , J- 

. -..i: . .JvT  '.  ^•^,'^l^^“*‘  r^.  « .. .. ..  ' .KO.-.  Ifi07-i 

.'  - ’M  .. i.  y;  iA-ij  ^ vfl'-XXol 


\. 


^ 


V.: 


r- 


w - . . )j.  - ir,,'if:a  • 
« ■ <■  _,  . 


jf 


••  - . , i . • < , u -* 


i •: 

n 

' ^ :::»  i 


0 -*  I 1 li 


' ( i ‘ r 


I ..  ■'■  d i V , r*  ( ”,  y j V . . if  O' 


t»:  ■ 


0- 


u^; 


u- 


4^  -•*  » 


I 


i'  I ~ H 


34. 

Equation  (46)  plus  two  times  equation  (47)  plus  equation  (48) 
gives 


H - Py1 

L J 


-E<^,  -<j>^)  (49) 


CH^ 


CO^ 


-2  s. 


Equation  (49)  is  tlie  equili'brium  equation  for  iaethtjae  coia- 
bust  ion. 


X.  Derivation  of  Energy  Equation  Case  III. 

The  relation  between  the  thermal  and  chemical 


energies  of  the  above  gas  mixture  is  derived  as  follows; 
The  chemic^iL  energy  liberated  at  absolute  zero  is 


Prom  CO 

reaction 

axHo 

<^0 

IT 

II 

II 

Oli^ 

IT 

n 

00^ 

dissociation 

IT 

H^O 

II 

IT 

o 

o 

II 

-cz(l-x)Htf 

) 

Prom 

c.e> 

) 

CH^ 

II 

H^O 

II 

-2cz{ l-y)H^ 

) 

combustion 

Adding  we 

have 

U -U,  ^ xH^  ' (a+g<-oz)f  yH-  • (b•^h^2oz)  f H-  ( -g  - cz) 

( -h  - 2cz)-f  czH^ 

U^=  thermal  energy  of  initial  mixture  j'ust  before  ignition. 
U = ’’  ’*  at  equilibrium  point. 


I . 


» / i T ' ; w i.*  ■ ^ ' ( 1 ►.  ) I - 


OiU  - ' { . ) :'i  i V JLftjUi 

- H» 

• X 


Y.' 


r • i * V V-.  i.  * * 


(:.:•)  !u.i 


r *•-. 


Ol  K 


! 


J 

9 

i 

1 


I U liO  I I 

t 

f>rfT  j 

i 

I I 

1 , 


I 


35 


The  thermal  energy  of  the  mixture  at  the  equilibrium  point  is 
found  as  follows; 

U = / 2TydT  ==  T(a+bTtfT"') 

■'o 

The  following  notation  will  be  used: 


« a,  + b,  T + f,  T^' 

mean 

specific 

heat 

for 

CO, 

a,^  + b^  -+  f ^T’" 

If 

n 

If 

II 

CO;. 

n 

n 

It 

II 

H;,0 

y^=a^+  b^T  ^-f^T" 

IT 

II 

IT 

n 

GH^ 

>;^=a^  + b^T  -f  f^T"" 

n 

If 

II 

n 

00^  present  at  equilibrium  point; 

(a+g+cz)x 

H^O  present  at  equilibrium  point: 

(b+ht2cz)y 
OH^  present; 
o(l-z) 

GO  + Oj.  t present: 

CO  = (a+g+oz)  (l-x) 

^ f 

OjL  - i-(b+h+2oz)  {l-y)+-|-(a-+g-H3z)  (l-x)t  |e-2cz-4^(ai-b^ 
Sum  -g-(b+h+2cz)  ( l-y)+^( a-^-g-foz)  (l-x)  +[i-2cz-i-( a+b)  + :^ 
H present  at  equilibrium  point; 

{b't'h-f2cz)  (l-y) 

Total  gas  present: 

m+i(g+h-c) -4^(a+g+02)x-|(b+h^2c:^y 


t 


/'  ? Ox'  v f 

:.l,  • r-i' 

. .*  < ■ 

••  tn'J’ 

ton- 

z-r^oJI-,  . 

nr.  DrjuDi 

{ '"  . 

r 1 r 

Ml.  . 1 

r 

V 

{ 

t; 

.’  1 M..  0<! 

1 ,t.  k i.\y 

W*  0 i f .it  ii\ 

i Iw  '.';; 

f-v  • si.  i l..\  4.1  ^ \ 

fUi  f).  ; 

■p<  t 

.(.  d ’ ,4 

V»« 

< r ••  n 

' t 

‘-•j,  .c 

;-  i « •'  ^ 

- .X 

■t 

}r 

r ' ••  I 

1 

T,(r ' ,-a. 

^ IT  n ..  (;• 

•>  • * ' 

fT  '• 

- V*- 

X.,t'  1 

f*  tl  *. 

•''  r , -V 

T w 3 

,.}»T 

r i - ' 

V U l»’  ? . 

u^-\q 

: - ■ i- ■ ' • v‘  ii  U-E 

'•(  J0?4 !C<  fO 

I 

f.-Di) 

( .V,t  «)  ^ 

•^.  Jl 

h.-.r.trti  ,0 

' ■ ';  ; - C / f rd  ) ’.  “ fraur& 

: J.  ■ nn  .i..-"o  jd  ^aatoiq;  H 

(i-X)  (-0^*  r^fC! ) 

: inm  .r'-J'-f’; 

v' ■•-  ) ■'  ^r- . :*3>  rj  )v-  f m 


- - ,:.r/  --  ■ -i= 

I 


36 


We  may  write  for  our  thermal  equation: 


( a-tgfcz)xHa,^^ 

-f(a+g+oz)|:(l-x)T(a,+b,T+f , T^ 
+ ( a-<-g+cz)xT  ( a^-»'bi.T-^f,,T’') 

+ (g+hr2cz)yH^ 

-i-(b-+ht2oz)-|-(l-y)T(  a,+b,Tf-f,  T*") 

— 

f-{b-f-ht2cz)  (l-y)T( a^-Ha^Tf T^) 
•+{bfhf2oz)yT(  aj+b5Ttf3T’^) 

■foz(H^  -2Ho.  ) 

+c(l-z)T(  a^-Ha^Tv-f^T*') 

-{gH  +hH  ) 

CO  n 

f(eff-2cz-|(a+bj|T(a,-/-b,T-#-f,  T^)  - 

(50) 


Combining  coefficients  and  transposing: 

(a+gtcz)x^<j.^-+-iT(a^+b,  T-#-f,  T^)  - T( a^b^T-ff 
+(bth^Zoz)ylHo,-^-iT(a,+b,Tf'f,T'^)  ^ T(a,-tb^T+f^T'') -T( a^tb^T+fjOfi 

L ^1-  J 5Q 

-c(l-z)T(a^v-b^Tffy  T’' )-f-|e+-ffa+i-(ggfh-/-oz^  T(a/b,  T+f,  T*) 
H'(bthi£cz)T(a,-+b^T-^fcT"’)-#-(gH^  -^hH.  )-cz(H^  -Ho,  ) -U. 

where  Uj_,  the  energy  of  the  original  mixture,  is  given  by  the 
following: 

(a+9+f)Tja,+  b,  T^+f,T*)+Mj.(a/VT.y--fe-Tl-) 
oil  ( a^+t,  T^tf^  )t  gl  j.(  a^+b:.VfiT^  HbSj  a T,+  f, 

The  following  are  expBSsions  for  the  partial  pressures  of  the 
various  constituents  and  the  equilibrium  constants# 
p ^ (a-fgj-ozjx  p p ^ (l3»h-i-£oz)y  p ^ P 


_ ( a+gfcz)  (l-x) 


m 


■D  - (bfhtEoz)  (1-y) 
" m' 


m‘ 


^Ai=-  m'^ 


p - . e-t^( a->-g-K}z)  (l-x)f^i•(b-^ht•2oz)  (l-y)  a•^b-f4oz)■p  e_^_ 

X fm'  y /^m' 

{—  = Tty  ?;  = 

(atgtcz)x  (b+htEoz)*’ y’'  „ 

tTr^ otl-z)  • (e^)‘'  “ 


^ j i,  , i4. -la:  fi,^ii‘7  ^ . 


I * • 
\ <*• 


) (.-.I)  ( :t>^^  ,.  )■; 

V •! 


■ i; 


^ r»’ 


{ _•  ;-  . • 

f ■ ■ 


( 


I 

I 


I ' 


£J  ) T \ f.)  ,X  -■  0':^ ' d>  t ) ■' 

* V B ^ \ 1 . ( ::  -i  > •■ 

r i , )Zyi:i/Vii  d)r 

I 

( .•’ (.  •'£)d^ 

} -i0^v‘‘.  C,  • 


fi'A/ 


; ’ 


)V( 


■-  ^4!:':-  .n)soT^ 


2:^  :'  4: 

■ c .: ) 


■ /•>  r.5 


0 > 


: ■!  i:.c  ^ > ■'‘•x'3  : 1' 

•1 

- ( '.s'  1'  ,0-  .O'"  ; ' ^v:(*.0-  " if) 

)'.  • C'T.'Sfc'rV  X .-ocf)  V 

• .'f% 

( ';  ? 1 .-3  ) £ ;r:o  il '.  v'*  'Ji  > - ■ ji' ^ 'i  )y  »» 

.1-  ’"n^  .jO 'r  ( "V  a,  )D  ( *;C'a’  >d ) ^ 

•V-  lo  , U 0't©::'.V 


iipvi 


. (Iir  A . ,:•  ,yd'( 

o'-ir,  • J , -.M  ^ -5 , V 


iit.-  ;.  s.\  *1.,  . . -v^l  #*44  -^nJ  .v  flo't  fexiir 


? (fry  S^^!£jL 
''h' 


: njL^*oJIot 

.V  0.  .)  '■  J 


.:  ■ /ICt.  : ;.|J1: 1 ■ i \ ■ ■ t»  fv.  e 


r." 


x7ai'!Co  •010X54^ 

( V f ♦ *»*  ■ • 1 ) 


ti)  ' 

fT 


0 ^ h i.  ':  , - ( •••X  4-  1 

. j . ...  .... 


.5 


■f-- 


4 » •»  ^ 

. J 


-T 


.1. 


( “'Os'-'.'  cl . ■/)  / 




■ H 


37. 

XI.  Reduction  of  Equations  &nd  Method  of  Solution  Case  III. 

Equations  (46),  (47),  (49)  and  (50)  are  the  four 
equations  whose  simultaneous  solution  gives  the  valies  of  x,  y, 
z,  and  T at  the  equilihrium  point. 

Substituting  the  numerical  values  of  the  specific 
heat  constants  and  expressing  in  terms  of  and  T according 
to  equation  (26)  v/e  have 


4.671 

4.571  log^K,^^=12|iOO 


->  —6  *. 

-5.3881  log^^T+1.3335'10  T-0. 08-10  T-13.1 

(51) 

-6.2621  log,^T+0. 2361  • lO'^Tf 0.0333*  ld^T+1.1 

(52) 


4.571  log,^;.  ^-^Z2^  + 5.6183  log,  T-0.0552*  10"t-0.0133*10”^T-22.4 

( a+g+o z ) X ^.1 . 34-1 . 3333  -10^ T+0 . 1 6 •IO^T'J 

-<■  ( b+  h+2oz ) y 1 . 72-0 . 2361  ♦ 10"^T  -0.067*  lO^T^'J 

- c(l-z)  |5.04+2.500*ld^:^  + ^+ffa+i(3g+h+c^|4.51-K).2778*10'^T] 

(54) 


, 121250g-H02100hj;cz. 21570  - Uv 

' T 


The  solution  of  the  above  four  equations  would  be 
much  simplified  if  the  value  of  one  of  the  variables  was  known. 

We  arrive  at  the  approximate  value  of  z by  the  following  method: 
The  maximum  value  of  z,  of  course,  is  1.  If  we  establish  a 
minimum  value  for  any  given  case,  then  the  vflue  of  z is  known 
to  be  between  this  minimum  value  and  1. 

When  z is  a minimum,  the  partial  pressure  of  the 
methane  constituent  at  equilibrium  is  a maximum.  Prom  the  equil- 
ibrium equation  we  have  at  any  temperature  T the  following: 


n i:  10  ‘ ^ ^ • 2a  ...  ■>  '. 

iiv  .1,  ^‘•■^  ) -'.-  \ . . ) , ' » , V J.  . fi  1. 

4.  ' • '‘<i.  : * . J .1.  t ' .L  ' X i>  1^  Vt  ‘ ^ ^ U i.'Tt.  \ • *W  ^<r*  i<f  * ' 

• •■lAi-J,  r X..*  w ;»,i'v  Cr  *r  ;..u  , 


•.  w ^ 


: .... . . ‘ j;,.  ; .:  Kcf;r<; ' 

,*■  iii  .;  /i  .'J  ’i  iin  ! '.\V  : hor  1.1- ;. 

• . , ...  • Ct/--.--  '>i«T‘'’:'5’l^i:t'  ijpo  Ow 


I X. 


I J.  ' 


01  ■£; 


.^OX  C 


. 0*  .I:::X 


. i:  ‘ r 


t'  I j.  ■ 

V'. 


30.1;  n 


.i.  OX'  « O’  • "*  - ^V>  J.  X> 


(■  • . 


0:.  ..  - OX  xva.  ^ 

, - jTl  - 

u.r*^  * 1 0.  . .*  .1  0 

t ‘ 

..  ./  -.  - ox  - . -■*  ' •■.(  J-v- 

- ' j .. 

ytjJJ  O M'O  J.  !;:i;::.^^SrJj:JixJ-.  ' 

..  •-.  .•  ,,  •••;<.  . 'io  ii  X'f^'IOG  ojr.X  ; V ;• 

■/ry  xo  o:V*  _i  .oei . .LlJp;  i:o  ^xixr* 

t*..j.O'  ijx  ..\’.:.»c  * ■ ■ ,i:  kj;  : ^ cvi  i'.:.j  os'- 
, r .7  *0  tfi.J  orix^ro  oc^l. 

„ 7Vi’'l  ■\.  - ,. 

, . ,:wVi ,’^.  c*  ?r.-0  fe.!.:!.;'/  "iirr-tn.lrj 

^0  -• 'r  7 i -.‘.X  ';«■*.  . . • .«/  , . '■>.  I . . 


Xt  ^ «4» 


..  ;.i.  7/L  . ’/ 


.iA  . .A 

. . t> 


L - K 


QJ 


.A  c .t 


. V - iKOt 


7a  w . r>ju^Xj  stroo --S^xfXdm 


:•••  .vA’dxXol  t’i'j  4'7-.;X  .i'A©..  ::.vJ  v.fX;  jo  :;v,ri  .-w  ar^io'dl 


\- 


r 


. A 


, ,t  ir 

"f' 


28. 

If  we  consider  that  there  is  no  dissociation  of 
the  CO3.  or  H^O;  that  is,  x =1,  and  y ^1,  then  the  partial  pressu 
of  these  two  constituents  will  he  a maximum.  Also,  the  0*.  par- 
tial pressure  will  he  a minimum  h ecause  more  0^  is  required  to 
completely  hum  the  CO  and  than  is  the  case  when  there  is 
dissociation.  We  may  further  decrease  the  0^.  partial  pressure 
at  equilihriurn  hy  furnishing  in  the  initial  mixture  only  the 
theoretical  amount  of  oxygen  required  for  complete  combustion. 
Under  these  conditions  the  right  hand  member  of  the  above  equa- 
tion is  a maximum  which  will  make  the  partial  pressure  of  the 
methane  a maximum. 

Substituting  the  values  for  the  partial  pressures 
with  X »1,  and  y -1,  we  have,  using  the  required  oxygen,  the 
following: 


V { a+gfcz)  (b+-h+2oz) 


From  which 
( 1-z) 


|(  atg-fcz)  ( 

" I 4c^ 


cz)  (b+h-hScz) 


X 

If 


By  putting  z 1 on  the  right  hand  side  of  the 
above  equation  we  get  an  approximate  value  of  z which  is  slightly 
smaller  than  the  value  indicated  by  the  equation. 


(1-z) 


■E 


(atg+c)  (b+h-f-2c)' 
4c'^^ 


1 


(55) 


From  equation  (55)  it  can  be  seen  that  z will  be  a 
minimum  if  c is  a minimum  for  a given  value  of  • 

Of  the  gases  used  in  gas  engines  the  methane  con- 
stituent is  the  lov/est  in  producer  gas.  An  average  producer 
gas  has  about  the  following  composition  by  volume: 


CO 

22  mols 

a 

CO^ 

7 mols 

8 " 

b 

61  ” 

f 

CH. 

2 ” 

c 

total 

100  mols 

Substituting  these  values  for  the  constants  in 
equation  (55)  we  have 


( 1-z) 


r(5i)(iE)"7 

L'  32  n/  J 


(56) 


e * .. 


- .!.  Iv‘  ■ 


1.' 


) 


c.  • 


•ij. . ^r*:  i i*v 


' c i 
G * t* 


- 1.  vv  ■■  i ' 

, J- ..  , ' ■ ' , ' I - ; w.  ' 'io  ,1,1 

oc  '.i-,  .t  Jl  ;.;'  L 0 ..r.^  .l 

’ '■ ‘.,1.  :'i  i- a IV  :•  XII>  . ■ X io' 

. ; .• -cTJ  i.Li  !.  ri«>i  •.  tuv  ."Cv/nic.: 

w < L.  “I'i  :.V -:;6  .1 -iJ  17  tV  . jioi 

'.  1 ■ : : ' - rJ  i >-:v^  *r;  ..  , i.-'ciXi:;!  » c.. 

• T 1' ?>t  - -'7;  . * .r. .j;,i  ..'  i • Li'J'o'iC'orij 

■ ;?i.,A-  s':  ;»  ■ 'J  r ' isOi-Ii 


X..U" 


,n  , ' » 


* r::ri7x^.  A ei  rtoiv 

. :■  -.  . a /.V  G4. 


o/J  , 


V « 

V 


: * l^Gci 

u frsv'  . X - jr'  » C - it  d^XV'- 

: yii'.vc  X lO-t 


! • y I * Cf*-  ' ■ ) 


^ { Tt  •: ) (;.,  o*  r^  )' 


I 

j 


rc'x'X 

( *i 

< , 

- • ) 


- : .r  c‘  ...  X ■;  f ^ 


L -V  i X r . ?.c*: 


i.'  ♦'rf  I 
r‘  , 


f;;  w vVs'Ji 


. ,, j .» ;f  i-  Tkv  i ,;f.  f 1;:  £ 


• £ v#XXji;:'\-: 

(;'-X) 


ou 


. : .<-5?  i.  I 
1 : 


L *■■■■—  i 

' X '■  ( ' i V XJ,„  V,:  t i 

iJil-  .:  vtv  aurjr.  w.  i* 


„u.‘*  .<j  ) J 


t < ■ i1 


•.j.  VV;  ' '-i  ;vW  ^0 

..O'.'  .,;  f.r- . Jrffri^'vt X't': 


^^4  V O 


” . Xi  ,:• 
w£.r"xTT  .£cwv:t 


" e 


iwJ"f_<w.vvy  •'..c.r  i IJ  j.;-! Jr 

f^7.  d O'.  • ( i ) ilQXi’JWJpft 


(r*) 


v\C»  I 


/ ( 


I ( 


J.  0 XI  *rrr7LlfxX;c 


‘jLs^d;;  t:a/(  ®r-3 


I ■ 

■ * 


39* 

From  equation  (53), log  = 13*4113  at  5600°F. 
absolute,  a liberal  maximum  temperature.  For  lov/er  temperatures 
ILp  increases. 

Solving  (56)  we  get  as  a minimum  value 
z - 1 - .A-,,..  0.9998^ 


V/e  can  therefore  take  z = 1 without  introducing  error. 

Putting  z = 1 and  substituting 

in  equations  (51)  and  (52)  we  have 

4.571  log^^  1^1 -X  (^QfM  gfh-c)  a+g-fc)xf(bth+c2) 

^1212^  -5.3881-log^^T'M.3335'l6"T-0.08'ld"T"'-  13.1 
^ 4i|72^  a . log,^T,J  ( 58) 

4.571  log^^  fl-y  g+h-o ) --s-fl  a+g+o  )x  +( t th+Eo ) J 

_ IQSIQQ  -6. 2631-log, ^Tt0.2361-ld^T+0. 0333*10^ T+1.1 


4- 


4.571 


[log,„T  +log,^P^  - log,^T J ( 59) 


Subtracting  (58)  from  (59)  we  have 


-3. 


-0. 8750-log  T-1. 0970-10  T 


4-0.1133  -lo‘Ti-14.2 


(60) 


The  following  notation  is  introduced; 
L-(atg+c)  n-(bi-ht2c) 

r=  ef-|(g+h-c)  s^  e +f +a+^(  Sgthtc) 


\ 30^.  , 


y 


I; 


)■  -I 


■ i. 


I; 


I 


M J 


f 

\ 


:c 


( • 


I 


r 


j. 


• r ■ 


.1 


I 


40. 


-5.3881-log,^Tfl.3333'10'^T  - O.OS-IO'T*'-  13.1 
f^(T)  1Q|12Q  -6. £631-log,^T^0.2361*10'^TK). 0333-10'^ T>1.1 

f^(T)  -0.8750.1og,^T-1.0972'l6"T+0.1133'l6^f-M4.20 


JZi 


f,  (T)-M^^tl«34  - 1.3333  *l6  Tt0.16*10’^T*' 


fj(T)-i9|i22  + 1.7E  - 0. 2361-10  T-0. 067-10  T 


f,(T) 


f,(T) 


'3 


4.51  + 0.2778'10  T 


12125Qg-M021001if21570Q  - U, 


From  (60)  we  get,  using  notation  given  atove: 

& 

For  any  given  temperature  0 is  fully  detemined. 

(61) 

Substituting  (61)  in  (58)  we  have 


= log"'  flitL2)l= 

x(  1-y)  |^4.57:y 


Gx 


log 

/o 


^ nOx 


__  f*(T)  . 1 n__m.l  Pi,  /,»v 

43^  ^2  '^^+2  (62) 


x(e-i)  ijj 

Substituting  equation  (61)  in  (54)  with  z 1,  we  have  the  follow- 
ing, using  the  new  notation: 

x'  j(G-l)L  f,(Tjjtx{L-f,{T)-(G-l)[s-f,(T)+f,(Tf|  + iiG-f^(r)| 
-js-fjCD+f^d)]  = 0 (63) 

To  solve  equations  (62)  and  (63)  proceed  as  follows; 
In  equation  (62)  assume  a probable  value  of  T sons- 
where  in  the  neighborhood  of  the  expected  result.  The  right 
hand  member  of  the  equation  and  the  value  of  G are  then  determined* 


41. 


Solve  for  x by  trial.  Plot  the  curve  of  (62)  with  x and  T as 
coordinates. 

In  equation  ( 63)  assume  values  of  T which  then 
reduces  the  equation  to  a quadratic  in  x which  can  be  solved  by 
the  quadratic  formula.  Then  plot  equation  (63)  with  (62).  The 
intersection  of  the  two  curves  gives  x and  T of  the  solution, 
y is  then  found  from  equation  (61). 

To  detemine  the  maximum  explosion  pressure  we  have 
P^m'T 


P = 


(64) 


XII.  Numerical  Example  Case  III. 

A gas  engine  uses  a gas  having  the  following  anal- 
ysis by  volume:  CO  20^,  CH^  1(^,  GO^  8^,  2^,  N^  20^. 

The  thecretical  amount  of  air  is  supplied.  Assume  10^  of  the 
products  of  combustion  left  in  the  clearance  space.  Initial 
pressure  and  temperature  of  charge  in  cylinder  14.7  Ib.per.sq. 
in.  absolute  and  190°P.  Compression  pressure  150  lb. per. 
sq. in. absolute.  Find  the  maximum  temperature  and  pressure  and 

the  extent  of  combustion  in  the  adiabatic  Otto  cycle. 

The  gas  mixture  in  the  cylinder  at  the  end  of  the 
suction  stroke  consists  of  the  initial  gas  plus  the  required 
air  plus  10^  of  the  products  of  combustion.  The  charge  in 

the  cylinder  has  the  following  composition: 

Mols  Mols 


CO 

4.00 

a 

0.62 

h 

H;. 

2.00 

b 

O- 

6.00 

e 

CH. 

1.00 

c 

23.10 

f 

GO^ 

1.38 

g 

Total  number  of  mols  - 37.10  = m. 


42 


The  constants  for  the  adiabatic  compression  equation  are 


34. 10'4. 51+4. 04+1. 38*6. 42+0. 62-5. 04 -=168.435 
2b^  = ( 32.10' 0.5556+2.0  *0.5 +5. 0-hl. 38 -3.5+0.62 -1.25) lO'^ 
-29.440-10'^ 

-f(-1.38‘0. 4840. 62*0.2)10'^ 

« -0.2692-10'^ 
mAR  ^37.10-1.985  ^ 73.643 
aj-mAR  - 242.078 

fn 

Substituting  the  numerical  values  for  the  constants,  equation 
(14)  reduces  to  the  following: 

log/,T^+^.72277j  T - 6839oJ  T^  = 3.15392 

(Brackets  indicate  logarithm  of  the  coefficient.) 

By  trial  substitution,  T^^  - 1230°F  abs. 

The  constants  for  the  energy  equation  are  as  follows: 


L=  6.38 

f^(T)=.  . 


; n =4.62; 

17945 

“TT" 


r = 5.50;  s=  34.98;  U^=234,252; 


Substituting  these  constants  in  equation  ( 63)  we  have 
x"  [6.38(G-l)f,  (T^+x[6.38  f,  (T) -(G-l)  ^34.98 

+4.62  G f jTjJ  - |34.98*f3(T)tlZ|i5j  :^0  (E) 

V/e  have  the  following  values  for  the  various  functions  of  T in 


equation  (63)  according  to  definitions  given  on  page  40. 


T 4700 

4800 

4900 

5000 

5100 

Log  G 0.9320 

0.9479 

0.9646 

0.9804 

0.9965 

G 8.551 

8.870 

9.217 

9.559 

9.920 

f<  (T) 84.411 

23.891 

23.398 

22.928 

22.481 

f;,  (T)22.041 

21.527 

21.030 

20.547 

20.082 

f,(T)  5.816 

6.844 

5.871 

5.899 

5.927 

It)  3.818 

3.738 

3.662 

3.589 

3.519 

' 


.L  J 


r 


i 


- i 


M 


t 

I 


f . 


j 


( 


.i.i 


) 


. .) 


A 


?*  ;j  0.  ■ : , { ;r  ) 


JC> 


■*a 


V... . . 

i . 


J.l 


J 


( ' I 


43. 


For  T 5000  equation  (E)  reduces  to 
X*-  -0*6935x-0.1677=^  0 
By  quadratic  fomula  x = 0.802 

We  have  from  equation  (B)  the  follov/iiqg  pairs  of  values: 

T 4700  4800  4900  5000  5100 

X 0.707  0.738  0.770  0.802  0.835 

For  the  second  relation  between  x and  T we  have  equation  (62) 
which  reduces  to  the  following  upon  the  substitution  of  the 
proper  constants: 

r5.50-3.19x-S.3l6x  li 

L L inm-rj 

For  T =*5000  the  value  of  the  right  hand  member  of  equation  (F) 
is  0.78662 

Solution  by  trial  gives  x = 0.800 

Similarly  we  get  from  equation  (F)  the  following  pairs  of  values: 
T 4900  5000  5100 

X 0.849  0.827  0.802 

The  intersection  of  the  graphs  of  equation  (E)  and  equation 
(F),  fig. 6,  page  44,  gives: 

T-  5050V  <>^3.  X =0.816 
For  T -5050,  G -9.751 
Then  equation  (61)  gives 
y-0.977 

Equation  ( 64)  gives 

P-  = 543.3  lb. per. sq. in. absolute 


= f^g^'^-ilog  1-0.16241  (?) 


u i . 


• >'  ; . 


. i ...  ■.  j 


I '•  ••  > V ^ V ^ ^ w • * » 

' V . ■ U',  . i' 


u 1 . . - .s  V I 


r ■ 


C ... 


• , r 


: ••  !'  . ; • ‘ .'V 


iU  ^ 


i.  i < . 


I ■!  L'  ■ V.  • ! 'j 


.-r.  I 


; ( L ) ■■ 


.'■'.ne 


r > 1 


t .1: 


( ■’.  I . '■  > 


• . r, 


r-  - 


45 


ZIII.  Effect  of  Loss  of  Heat  Luring  CornLustion  on  the  Maxim'um 
Temperature  and  Pressure  and  the  Extent  of  Combustion 
in  Case  III. 


To  show  the  loss  of  heat  by  radiation,  conduction, 
etc.,  during  the  combustion  phase  on  the  values  of  x,  y,  and  T, 
various  percentages  of  the  heat  of  combustion  ^all  be  assumed 
as  lost  and  the  corresponding  values  of  x,  y,  and  T calculated. 

The  only  change  necessary  in  the  equation  for  the 
adiabatic  case  is  the  subtraction  of  the  desired  amount  of  heat 
from  the  left  hand  member  of  equation  (50).  This  results  in 
an  additive  term  in  the  numerator  of  f^ (T) .TAe  equilibrium  equa- 
tion is  not  affected. 

Let  H=  total  heat  of  combustion  of  gaseous  mixture  at  constant 
volume  and  6E^P. 

100  K percent  of  heat  of  combustion  lost. 

Equation  (63)  then  becomes: 

f,(T^+x|L  f,(T)-(G-l)[s-fj(T) +f^(T)t^]+nG-f^(T)j 

-js  f3(T)tf^(TH^j  = 0 


The  above  equation  with  numerical  values  for  the 
coefficient^  that  of  x being  equal  to  1,  can  be  obtained  from 
the  simplified  equation  in  the  adiabatic  case  for  a given  temp- 
erature by  subtracting  from  the  known  term  of  the  quadratic 


KH 


T[;iG-i)L-"f7nrr| 

X the  value  of  this  fraction  multiplied  by  (0-1 ) 


and  from  the  coefficient  of 


equation  the  value 
thii 

If  values  of  K that  are  multiples  of  each  other  are 


chosen, the  quadratic  equation  in  x and  T can  be  immediately 


1 


r j •>/ 

'TV 

’ . 1 
X A ^ ^ 4 • .JL  s 

* / , 

.*  ■ w r»  ^ '; 

.0  i i;  ' 

\ 

» 

r** 

■*4  '•  I *'  .A.  1 r 

4i.S'  V-  . ■ X . *lv* 

rVO./^t> 

' \ • M . I*  , *■  1 1 ' 

’ or.',  .*  r i 


Jt^i^L-i  -.  t;i>rX;.lf  r il: '.  ;■  io  « ■>■,  \5r,J  :...  ^ a^joi’i.-v 

* -''':.■  I',  ol -t  '■  i;:r.  , 1&  ,c;c:;.C.v  ’;■  I^..;y';,'J‘C'^ oJt  c.. 

J ‘xo  - ).  i,;,....:  :.'  .'..J'l.l  V-  oru:«  o UV'  f5';X 


:.(• 


{ .*.  - * i '-'f.w 


0 X *..•  i L < **  ' *D  •••  i 1.'  V X ^ - X t 


...I  .,' u L ! .i<\  i.  i . • . \>v . / ii  A J C 1. 1 '.)  IT'  '» 0 f.’JOl.'J  i;  k.  iOJ.  Ollk'  i'tO*;  il 

- ::■  i-  '•;.  .(  ) 1 'to  '-C^ -■*;.  . a.,;'.  oi-J^  r *'f- 

• y ^ V 1 

‘ .L  • ^.  Aj  \ ^ ' i,»  ^ * ...  i.’  - ,t.  ^ -,►/  *i . * K ^.  ■ u ^ w 0 ^ ^ 0 - 


•vj.u 

c -I  ri*^' xu 


Ij  J jt  t!i  if-^  .i  {•  '> 


M 


> V , :, 

« ' 

Ir.C  -l  L..  ..'.)  »j;  : ;.o  I"'  -••  .j'i,  ^ OCI 

.'  .ii.  . C i.  ■ t y;  <>  (f'iv  ) f'C  C'^’  'ir''<^  H 

♦ 1 ■ 


> (■'it. 


f{-]  •..  . ,.  / /■r’l 

\ -i  / .^  • J I -s  . } 


UT'-v  .,  (■:  ).1 


0 - ' I,  :^r{ 


X^K.  !.*.•::/•  uiTJ.v/  rv. ^t.<,.'.  f,-..T 

, V 

. ‘ T r / , ' > <*  ‘ \ , .'  t *z 


i ri*l  _ X.- J o . 

- iv  .' 

run)  , X Ow  , 

w‘  i':w7‘ 

01  •■  :>  Li  j.~  ' c."^ 

ivl  J -i  .3  c'f/j 

v,.iw* 

.i'  r-xf-s;' 

to  V nt  '.  f . 

.'I-  t.£'. 

i.'.u'l  !, 

(l~ 

i ) . ■ ; c /;  iw’  i X 

<•  A. 

tu  7 

^ »ji « ■; 

:X  0 X /aV 

r.''j  ■<..').  :.'■. ‘it  '-.'/pfj- li.  uiti  acw  -j 


■ V / 


. , /'i  J'  J Qiiv  . X"‘  O'..  »-'-■  I 

'•  ■ '’' ' .i 

.J  ' A-  -**•)...  , ', 


iw  'x.Oau;.. 


•IJ  aimut  ••■  ■•  ■ ---5«ea(Beaf5»v--'^.;?£=:  -..r 


46. 

written  down  from  the  equation  of  the  adiabatic  case  after 
the  correction  terms  have  been  calculated  for  the  first  value 
of  K. 

The  following  values  of  K are  therefore  chosen: 

Case  No.  0 1 2 3 4 

0 2.5  5.0  7.5  10.0 

H 1,038,070  from  analysis,  page  41,  and  heating  values 
given  in  Appendix  B. 

For  5000°'  and  Case  1; 

'm 1 _ 0.026  • 1038070 ^ 

tr‘  ((J-l)L  - f,  (DJ  5000  • 8.559  - 6.38  - 22.928  " 

j(G-l)  = 8.559  ' 0.00415  ^ 0.0355 

For  Case  0: 

0.5935X  - 0.1677  = 0 
For  Case  1: 

X*'  - 0.6290X  - 0.1718  =0 
For  Case  2: 

X*"  - 0.6645X  - 0.1760  ^0 
For  Case  3: 

x^  - 0.7000X  - 0.1801-0 
For  Case  4: 

x’'  - 0.7355X  - 0.1843  -0 

For  the  other  temperatures  we  deduce  similar  sets 

of  equations. 

Solutions  of  the  various  quadratic  equations 
give  the  following  table  of  values  for  x; 


* 


•'“iw  . . •:  ► I-  Cjt'iVI 


\ 


} 


V* 


.'I 


- a 


t 


. » 


J " 


I 


1 


< • • j J.  * - I 


1)1 


o 


f. 


'fO‘j 


47. 


T 

4700 

4800 

4900 

5000 

5100 

Case 

Uo.  0 

0.707 

0.738 

0.770 

0.802 

0.835 

Case 

No.  1 

0.738 

0.770 

0.802 

0.835 

0.868 

Case 

No.  2 

0.772 

0.802 

0.834 

0.867 

0.901 

Case 

No.  3 

0.802 

0.834 

0.866 

0.900 

0.934 

Case 

No.  4 

0.833 

0.866 

0.899 

0.933 

e.967 

The  following 

pairs  of 

values 

satisfy  the  equil 

ibrium  equation  (P)  , 

page  43*. 

T 

4000 

4300 

4600 

4900 

5000  5100 

X 

0.970 

0.945 

0.905 

0.849 

0.827  0.802 

Plotting  the  above  values  we  have  the  chart,  fig.  7,  shown  on 
page  (48).  The  lines  for  heat  losses  from  15^  to  ZQfJo  were 
obtained  by  extrapolation. 

The  following  values  of  x and  T are  obtained  from 
the  chart.  The  values  bf  y are  then  calculated  by  means  of 


equation  61  and  values  of  P 
% loss  of  heat:  0 5^-/- 

from  equation 

10  15 

64/ 

20 

25 

30 

T '^I.abs. 

5050 

4930 

4810 

4680 

4550 

4410 

4260 

X 

0.816 

0;.843 

0.868 

0.892 

0.913 

0.932 

0.949 

y 

0.977 

0.980 

0.983 

0.986 

0.988 

0.990 

0.993 

P#/sq. in.abs. 

543 

529 

516 

500 

485 

468 

452 

The  curves  on  page  49  give  a graphic^  representation  of  the 
data  given  in  the  above  table.  Prom  this  curve  sheet  it  can 
be  seen  that  x and  y increase  with  heat  lost  during  the  explo- 
sion phase  while  T and  P decrease. 


X 


0.36 


ii!i 

-p.- 

\ ‘ 

I’i'^ 

':.1 

. 

' 

' , 

' 

i 

■ .r' 

-pr  - ‘ 

% 

y-^ 

i 

\ 

e* 

' 

mM-i: 


[4700:^  s/0 


460Oxi  4m 


of  Heat 


Dun/1^ 


the  Mmimurn 


Expios  f on  on 


Te imperature  and  Pr^ssute 


the  €\ient  of  Corpbu^^tionj^ 


j/fe/j-ce/it  Z.05S 


■\V.)  V 


■'tH-l-p 


.1.. 


p- 


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— • Mii  -Li- 


X. 


!*i. 


ti|:rrl' 


i4- 


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■ X.. 


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! 

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i > 

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tL.  i > ' ' 


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:jj'; 


50. 


ZrV.  Effect  of  Excess  Air  on  the  Maximum  Temperature  and  the 
Extent  of  Combustion  in  Case  III. 

To  show  the  effect  of  excess  air  on  the  values  of 
X,  y,  and  T,  the  preceding  problem  will  be  recalculated  using 
various  amounts  of  excess  air.  It  will  be  assumed  that  the 
compression  temperature  is  not  affected  by  the  excess  air. 

Let  100  B ^percent  excess  air. 

To  the  analysis  of  the  charge  in  the  cylinder  it  is  necessary 
to  add  to  e,  the  amount  of  oxygen,  Ee  and  to  f,,the  amount 
3.8Ee.  In  our  general  notation  the  following  changes  are  made: 
r is  increased  by  the  amount  Ee 

S ” n Ti  If  n 4,8Ee 

f^(T)  is  decreased  by  the  amount  4.8Ee  T«.( a, T^. ^f,  Tp 


This  change  results  in  subtracting  from  the  laiown 
term  of  the  reduced  form  of  equation  (B)  the  expression 
4.8e  fj(Tj  - 

CGrirr’X'  f/n?') 


E 


and  from  the  coefficient  of  x the  above  expression  multiplied 
by  ( C -1 ) . 

Having  found  the  value  of  the  quantity  inside  the 
brackets  of  the  above  expression  for  any  given  temperature  ^d 
having  the  reduced  quadratic  equation  from  the  solution  for  zero 
excess  air, the  value  of  x is  easily  determined  for  any  percent 
of  excess  air. 


61 


V/e  have  from  the  solutions  the  folloiving  values 

of  x: 

Temp.°P  Percent  excess  air 


absolute 

0 

10 

20 

30 

40 

4600 

0.678 

0.682 

0.689 

0.694 

0.700 

4800 

0.738 

0.803 

0.867 

0.935 

1.002 

5000 

0.802 

0.873 

0.945 

1.017 

1.090 

The  only  change  in  the  equilihrium  equation  (62) 


as  given  on  page  43  is  the  addition  of  the  quantity  Ee  to  r. 
Making  this  change  the  following  values  are  calculated; 

Temp • °P 


absolute 

Percent  excess  air 

0 

10 

20 

30 

40 

4300 

0.945 

0.968 

0.977 

0.981 

0.983 

4600 

0.905 

0.934 

0.948 

0.955 

0.960 

4900 

0.849 

0.881 

0.900 

0.912 

0.921 

5100 

0.802 

0.835 

0.856 

0.872 

0.883 

Plotting 

the  values 

given  in  the  above 

tables  we 

have  the 
solutions 

chart,  fig.  8,  on  page 
1 are  obtained: 

52,  from  which  the 

following 

^ excess 
T abs. 

air  0 

10 

20 

30 

40 

5050 

4975 

4890 

4790 

4680 

X 

0.816 

0.866 

0.902 

0.930 

0.952 

y 

0.977 

0.984 

0.989 

0.992 

0.993 

P#/sq.in. 

abs.  543 

537 

530 

522 

512 

The  values  of  y are 

calculated 

from  equation  (61) 

and  the  values  of  P from  equation  (64)  as  before* 

The  above  table  is  presented  in  graphic^,  form  on 


page  53,  fig.  8 a 


t-  : - 


■j’}  a> 

Vi  n i 


r 


Of.  1 

' rO<  i'V> 


a 


■i  * 

k*'a 


«4 


i 


O'  ’.•  . . 

O^L-  0;/ 


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..:»i  ; •',  iu 

' ' V'.  % . , > : * ' 

• • ii  ■ < ’ f - '*.■  * -*- 

. ..■  , i.i..'  7.  , \ 

w'.  V , ‘ ' C.  ■•  . .’•if:  • ' ♦ 

!,'  .. ,'  ... '1.  V \,  i .'■'.  \ 4’ 

. I t: . J.'vi ..;  Up©  uiCA.^. 

';  '1.  ■ .i  ■/ u^:  *'■ 


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035 


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rj 


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f ^ r 

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o> 

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P . . 

a . ^ 

£? 


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t ‘ ".■■■■; 

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: St 

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L 


til' 
• ■'■  i 


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rv* 


/"> 

<v 


•>?  U.:-N‘.-  . '-'  K 


I 


': 'Mm IjJiy i : uiiiiiL±iiJ-^4iiL  ^ 


' '.O' 


54 


The  effect  of  excess  air  on  maxiraum  explosion 
conditions  is  the  same  as  the  effect  of  heat  lEt  by  radiation, 
conduction,  etc. , namely,  to  reduce  the  amount  of  dissociation. 

In  the  case  of  the  actual  gas  engine  where  both  factors  are 
present  we  may  conclude  that  there  is  very  little  dissociation 
of  and  11^,0 . 

ZV*  Extension  of  Analysis  to  Cover  Gaseous  Combustions  Involv- 
ing the  Hydrocarbons  C^H^,  and  C^H^. 

We  can  extend  this  analysis  to  gaseous  mixtures 
containing  such  hydrocarbons  as  C^H^,  C^H^,  C^H^,  C^H^ , etc., 
if  we  assume  that  the  combustion  of  these  hydrocarbons  is  com- 
plete and  that  the  reaction  proceeds  in  one  direction  only.  In 
other  words,  the  final  products  of  combustion  are  00^  and  Hj,0 
which  establish  an  equilibrium  with  respect  to  CO,  and  0^ 
without  regard  to  the  nature  of  the  original  hydrocarbon  from 
the  combustion  of  which  the  CO^  and  H were  formed.  The  basis 
for  this  assumption  is  the  work  of  Professor  W.  A.  Bone  and 
others*  on  the  combustion  of  C^H^  and  C^H^  each  in  the  presence 
of  its  own  volume  of  oxygen.  In  the  cases  v/here  the  initial 
pressure  was  sufficient  for  detonation  the  products  of  combustion 
of  both  the  mixtures  C^H^fO^and  were  CH^,  CO,  CO^  , Hj_, 

and  H^O.  According  to  the  previous  discussion,  if  sufficient 
oxygen  had  been  present  the  methane  would  have  burned  completely 
so  that  the  final  products  would  be  CO,  CO^,  H and  HJD.  Since 
there  is  always  sufficient  oxygen  present  for  complete  combustion 

*Phil  Trans  Roy  Soc  v215A  (1915) 


55 


in  gas  engine  mixtures,  we  may  accept  the  assumption  of  the 
complete  combustion  of  the  hydrocarbons  as  being  very  close  to 
the  truth  in  gas  engine  explosions. 

It  has  been  shown  previously  (page  6 ) that  lack 
of  specific  heat  data  at  high  temperatures  for  the  hydrocarbons 
introduces  no  serious  serror  in  the  calculation  of  the  compres- 
sion temperature.  We  are  not  concerned  with  the  specific  heat 
data  of  the  hydrocarbons  above  the  compression  temperature  be- 
cause in  the  energy  eq.uation  we  consider  the  thermal  energy 

of  the  final  products  which  with  the  above  assumption  will  not 

energy  of  the  initial  mixture  which  is  at 
contain  any  hydrocarbons  and  the  thermal/compression  temperature. 

With  the  assumption  of  the  complete  combustion  of  the  hydrocar- 
bons the  conditions  at  the  maximum  explosion  point  of  any 
gaseous  mixture  ordinarily  met  in  practise  can  be  calculated  by 
the  methods  of  the  preceding  pages. 


* 


•»Qi  to  e^*r  <r(^«co4  \/ft^  e^  ^a^'SJJrln  ftjcr^^o  ujbs 

04"  &acIo  x^«»v  ’ai^orf  • ttap/fii>oO’itv;a  •ti-  oi^Jt4fe»iJrfiaco  i^nXqjljoo 


«or.:.  dd^l^re  u«%  rtt  :#it4 


r.  -wt'  ■■'  ;:  \ - ■* 

‘ 3tl  U ( 3 ti  oooU  mixl^'41.  '^  * ^ ^ 

Baodxoco^fr^  eriJ^  loTc  xi:- -'xciw»4  cijid  4-i  «4«Ji  4^od  ol^-toeq^  io 

i 

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' • ^ • • 

4a©  I QlsXa&cru  itdUiiXtw  i^&niat^ffco  4ML.  •*  hcIb 

-®<f  i.iyT4|‘i6qiKe4  nolGseir^go  ©xl4  ©votf©  GnO(/t©oo'$^'i<t  ftd&  XQ 


V" '.' 


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Ix^mi*a4  ©dJ  ■xa^Iono©  ©©  nc44«np©  ^^-itno'orf^  td  f©i;©c 


i 

'■  i 


4cn  Iltv.'*  ffCi4*^fc©i>  ©rotfa  ©44  d4iiai  doJtrtw  84ofrX>o^q  XsnlY  od4  Iko 
4«  ai  liolf^w  gtp^tjUm  X*iUini  frfu  to  \;3ioa6 
oi«j4ai©qmt'4  noie«©*i*ncoo\l3o;*iad4  ©iCr  i)na  ttnoji9oox5^  w>  nJt©4aot  , 

-inooxtiii  ©44.  to  acl48^ooioe  tO  noi4.(aiyi«©  od4  44jtW 

to  ittkoq  aoliiolaxt  s-asil-^a  ©44  4©  im©i4iMoo  ©xU  ©^Ou 


jbeJcXuoXi  O od  ii»io  ©e44o#^q  «i  4$w  niiiiuxlAio  ^Ofiuejs 

, . , .30^^  snXfceoouq  od4  flJyofI4oa|i  ©44 


56. 

Appendix  A 
Specific  Heats 

The  following  is  from  manuscript  by  Prof.  G.  A. 
Goodenough,  and  gives  specific  heat  equations  for  0*_,  CO, 
air,  H^O , CO^  and  CH^. 

”In  the  analysis  of  the  processes  occurring  in  the 
internal  combustion  engine  an  accurate  knowledge  of  the  specific 
heats  of  the  various  components  of  the  gas  mixture  is  of  first 
importance*  The  specific  heats  are  required  (1)  in  the  deter- 
mination of  the  variation  of  the  heat  of  combustion  of  a fuel 
with  the  temperature;  (2)  in  the  calculation  of  the  temperature 
' attained  v/hen  the  fuel  mixture  is  burned;  (S)  in  the  establishment 
of  the  equilibrium  equations  for  the  various  reactions  that  re- 
ceive consideration. 

At  the  present  time  it  cannot  be  said  that  our 
knowledge  of  the  specific  heats  of  gases,  eppecially  at  very 
high  temperatures , is  satisfactory.  The  results  obtained  from 
experiments  are  discordant.  Any  expression  for  the  specific 
heat  of  a gas  must  be  regarded  as  conjectured  and  subject  to 
change  as  more  reliable  experimental  data  are  obtained. 

Experimental  methods:  Two  principal  methods  have 

been  used  in  the  deteimiination  of  the  specific  heats  of  gases. 

(l)  The  constant  pressure  method  in  which  the  gas  under  con- 
stant atmospheric  pressure  is  made  to  flow  through  a heater 
and  then  through  a calorimeter  where  it  is  cooled.  A comparison 


I 


' i 

\ • i ^ , • -..in  fiXJ.  . • X U:e  I-nr*' 


fi 

[1 


r;-‘. . 


’’  c. 


0 


‘•p. . : 


I t - 

- .!.  'J 


J O ..  . ' L-t  \i 


k .ft  0 


u 


{'. 


/ 


i ' “ ‘ .■ 


t .'HO 


, r 

j . -I. 


0 


r 


l- 


. u 


jj 


. -t' 


:-i 


'..  j.' 


-I.  i j. 


— ... 


, . I .. 

' * ^ .1. 


\ 

I 


OJ  U 


V/ 


I ' ; 


f 


i. 


w 


■1' 


rx-.'. 


■du 


X/j'  ; 


; 

s 


(i ' 


H ' 

I'i' 


67. 

of  the  temperature  change  with  the  heat  rejected  gives  the 
specific  heat.  This  method  was  used  by  Regnault  and  subse- 
quently by  Joly,  Holborn  and  Henning,  and  others.  It  is  ap- 
plicable at  moderate  temperatures.  (2)  The  es^losion  method 
used  by  Langen  and  by  Pier.  A combustible  mixture  of  gases  in 
a closed  vessel  is  ignited  and  the  rise  of  pressure  is  determined. 
The  change  of  temperature  is  calculated  from  the  pressures,  and 
assuming  that  the  chemical  energy  of  the  original  mixture  is  all 
transformed  into  thermal  energy,  the  mean  specific  heat  of  the 
gas  mixture  for  the  temperature  range  involved  is  readily  com- 
puted. The  explosion  method  is  the  only  one  applicable  at  very 
high  temperatures. 

A comparison  of  the  results  obtained  from  the  two 
methods  shows  that  the  explosion  experiments  give  specific  heat 
values  somewhat  higher  than  those  obtained  from  the  constant 
pressure  experiments. 

A critical  discussion  of  experimental  methods  and 
results  may  be  found  in  the  Report  of  the  British  Association 
Committee  on  Gaseous  Explosions.  This  is  published  in  Clerks 
”Gas,  Petrol,  and  Oil  Engines,”  Vol.l. 

The  Matoiaic  Gases.  Experiments  on  the  diatomic 
gases  as  nitrogen,  oxygen,  CO,  and  air,  appear  to  establish  two 
facts. 

1.  The  specific  heat  is  a linear  function  of  the  temperature. 

2.  The  specific  heat  per  mol  is  nearly  the  same  for  all  these 
gases. 


■1  ti. 


. (.'-1  « 


- ■'/  V'f  ^Ct^!  fl.  Vf  :;vV  . . oi;  wi’*’iD3f  „ 

-■*£•  ~.c  • ■ ‘:'’C  ■’  ’ • :!•  , fxi.\  , .f'»“i.  v.. 


< 


■ ■ '•'  ( '-' ) • Jt- ’i. 'll ‘i’.icJ  ' r:  :,i., J’ii  •. 

■ ■- ■ ‘'.i.  ' -X.'.^.  J.:  ...  ; 4 . /;u.  yo^ii;-..'  *’,d  = 

J i*  * . ^ J . 1 ’.' ■ V S 

i»  t '^  - ■ 5'-  • c^;”^  k!  .'C  .*,.J<%  Ci.v  Xv-i.1  tJ  j.  V !..'.*  t'  5 i.w  iO  ; !.  I J<  t; 

-<  a.v;  qtl  . a'  .'.  ...i  2 iv.  vr  - i.i  x i ,..j  : 

e.  X . .u\  • JLL  7driJ  • - X'.  I 

-r;Ci.  >^i*:  •.:  , ...V.  c-vnl  . :ixji  f-i  x ‘Xu 

ov  J ;;  oX.- , : xi  .'■  • ;.r.c  ; : < ^.X  oc;;^Su;  .-XX  .?i  " d-a^'  . ‘jOCUlo 

c 'vX  r*C'.  : . . ri  J .-o':  &i'x  1',  .too  : r;  ‘ *0  "A  s, 

• w' X : / vj v_t'.  ?"7l  . bj...  i'j.  i:cXuO'i,p:o  v.'’ : .i  3t  X ^vt:  Hr:  i>X>oHXvu. 

tr  '■. . 0 ucOf*  ;ii  'w  • ;.i  X :\ 

.»-  ..  J . :c>r:.X.  i .-jO ''.o 

X.':  ?.  l',,i‘'!'^  cn  J.  V.  r»v  ^'ii‘<,.  .iC  .f. otii..  j *y  .x^o.i.X.1 '.' j Xi. 

,'^.01  :.■  Ov.:*rr-.-.  ,‘OlXf'iu  pXX  xv  tJX  iX  .‘x  i;  >'i  ei  v;-n  ..Ju;:iC'i:'- 

fc.  .*  i A JI'ijxrfT  ti  *:jO  A • -' ' -<  .'i  ..  X C fft  OsXX  A..^.:^pO, 

. t.  Co'.'  .t,rv: IX  J _JX:r-  .loilei-  . axsO” 

'J-  - ''  V If!^' ' o "'  .f.  CL  F^ti  L 


J ':i.^  r*/' 


V.-X  /LniX  A.J.'.e  L>J  h:n  ,1.  , , aox-o'^Xx/i  ti  > 

: . •■  51 . .J  X >:  !.'  V 

orii  *t.:  riv  xXo."j:i'r  -i.c^j^TX!  l:  ?i  i.  I XX  oo<:>;?2  ortT  .1 

c:n;:x  n:  ' mcX  cH*  rS-:  i lo;n  *iy  « ol'IXot>-;2  oXl?  -X 

' . ■'.r  * ' . ♦ ij  *j  ^ 

•>*’  '*  . 

» .7 

> . . • U-'  • - ■ 


58 


The  equations  representing  the  experiments  of  Lan- 
gen,  Pier,  and  Holhorn  end  Henning  are  respectively, 

Langen  +0.0006t 

Pier  y --  6.9 -|-0.00045t 

* ty) 

Holhorn  and 

Henning  6*58 +0.00053£t 

In  these  expressions  t denotes  the  temperature  on 
the  C scale,  and  denotes  the  mean  specific  heat  from  0®-  t^C . 
Expressions  for  the  instantaneous  specific  heat  are  obtained  by 
doubling  the  coefficient  of  t.  This  we  obtain 

/p  = 6.8  + 0.001£t 

6.9  +0.0009t 
Vp=6.58-^  0.001064t 

Lewis  and  Randall*  propose  as  a compromise  the  formula 
6.50-t-0.0010T 

in  which  T denotes  absolute  temperatures  on  the  C.  scale. 

The  proceeding  formulas  apply  to  all  the  common 
diatomic  gases  except  hydrogen.  Eor  hydrogen  Pier  gives  the 
mean  specific  heat 

ifp  - 6.7+  0.00045t 

• m 

whence 

^p=  6.7  + 0.0009t 

That  is.  Pier  makes  the  variation  with  temperatures  the  same  as 
for  the  other  gases,  but  tikes  6.7  instead  of  6.9  for  the  abso- 
lute term.  Lewis  and  Randall  propose  for  the  specific  heat  of 

hydrogen  the  equation 
2Tp  = 6.5  +0.0009t 

to  Chem  Soa  vol  34.  1912  


59. 

which  agrees  quite  closely  with  Pier’s  equation. 

The  formulas  of  Lewis  and  Pandall  may  he  accepted 
as  a fair  compromise  of  the  conflicting  experimental  data.  To 
transform  the  formulas  to  the  P. scale  we  multiply  the  coeffi- 
cients of  T hy  5/9.  If  we  let  T/lOOO  = 0,  in  v/hich  T denotes 
absolute  temperatures  on  the  P. scale  the  formulas  become: 

Por  0^,  00  and  air,  Por 

6.5^5/9  9 (1)  yp  = 6.5+i9  ( E) 

Carbon  Dioxide:  The  experiments  on  the  specific 

heat  of  carbon  dioxide  may  be  classified  as  follows: 

1.  The  earlier  experiments  of  Regnault  and  Wiedemann  at  low 
temperatures.  These  have  been  supplemented  by  the  experiments 
of  Joly  and  Sv/ann  also  at  lov;  temperatures. 

2.  The  experiments  of  Holborn  and  Austin  extending  to  about 
800°  C and  the  later  experiments  of  Holborn  and  Henning  which 
covered  the  range  0®  to  1400°  C. 

3.  The  explasion  experiments :0f  these  the  experiments  of 
Mallard  and  Le  Chatelier  may  be  left  out  of  consideration  as  the 
results  are  obviously  inexact.  Langen  and  Pier  have  also  used 
the  explosion  method;  the  latter’s  experiments  extending  to 
2100°  0,  There  is  little  doubt  that  Pier's  results  should  be 
given  considerable  weight. 

The  following  are  some  of  the  formulas  that  have 
been  proposed.  In  these  denotes  th  mean  specific  heat 
per  mol  between  0°  0 and  t°  0. 


r 


r 


if.;' 


. vTiJr.:  .-  " V. :.  ■■ 

‘ vr-t.'i.i: 

:'i  /;  . 


• i ■ ■**'  -tw  Tii  t’  IlC  6u  3 

. /...,  .»i  .;-.T  xO  .:;  i'.;- V,.'  ■■'  e;‘fx 

iw  i‘ ' 1. c 1 a r_. 
< ::x  ca . ' o.r  •,,  'Xc-;.'*:  j’!  oui’ 


{-  ■•  r,":  J*-!  ..T 


■ 1 ’^c  e i 


I k V 


L 


i .) 


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60. 

Holt) or n and  Austin, 

^ = 8.923  + 3.046 -ld^t-0. 735 -10^  t' 

Holfcorn  and  Henning, 

4 = 8.84  +3.267'ld^t-0.792-ld^t" 

Langen, 

4 ^8.7  + 0.0026t 

'‘m 

Pier, 

4-8.79  +3.3 -10"^t-0. 95  *10'^"+  O.l-lO^t^ 

Lev/is  and  Randsll  have  proposed  the  following  formula  "based  on 
Pier's  results  at  high  temperatures  and  Holhorn  and  Henning's 
results  at  the  lov/er  temperatures: 

>^-7.0 + 7.1‘ld^T  - 1.86'IO^t"' 

In  this  formula  T denotes  absolute  temperature  on  the  C scale 
and  Yf,  is  the  instantaneous  specific  heat  per  mol. 

A comparison  of  some  of  the  proposed  equations  is 
shorn  in  fig.  9,  page  61.  The  points  of  temperatures  500-2000® 
are  obtained  from  Holborn  and  Austin's  formula,  the  points  for 
the  higher  temperatures  from  Pier's  formula.  It  v/ill  be  ob- 
served that  at  the  lower  temperatures  the  increase  of  specific 
heat  with  the  temperature  is  quite  marked,  but  at  the  higher 
temperature  the  rate  of  increase  becomes  smaller.  This  decrease 
in  the  slope  of  the  Cp  curve  is  clearly  indicated  by  the  experi- 
ments of  Holborn  and  Austin,  and  Holborn  and  Henning;  and  the 
indication  is  strengthened  by  Pier's  experiments,  which  give 
points  on  a curve  continuous  with  the  Holborn  and  Austin  curve. 
Langen' s formula  represented  by  the  straight  line  gives  reason- 
able values  up  to  2000®  but  for  higher  temperatures  cannot  be 


I' 


page  6f 


lOOO  2000  3000  ^OOO 


\ d 


'tlli 


'■•  I 


accepted. 


6E. 

For  convenience  in  subsequent  applications,  the 
equation  should  be  a polynomial  in  ascending  pov/ers 

of  T,  and  if  possible  of  a degree  not  higher  than  the  second. 

Our  problem  then  is  to  frame  a second  degree  equation  that  will 
represent  with  fair  accuracy  the  experimental  results  at  the 
lower  temperatures  and  also  Pier’s  experiments  at  the  higher 
temperatures.  Obviously  a second  degree  equation  v;ill  give  a 
maximum  at  some  value  of  T,  and  at  higher  values  of  T the  cal- 
culated values  of  C p will  begin  to  decrease.  There  is  no  exper- 
imental evidence  that  the  specific  heat  attains  a maximum  and 
then  decreases;  hence, at  temperatures  exceeding  the  temperature 
that  gives  the  maximum  specific  heat,  the  values  given  by  the 
second  degree  equation  must  be  regarded  as  approximate. 

Holborn  and  Austin’s  equation  gives  a maximum  for 
Gp  at  a temperature  of  E883°  absolute  F;  from  Holborn  and  Henning’s 
equation  the  temperature  is  2982°.  For  temperatures  above 
3000°  values  of  C p calculated  from  these  equations  decrease  rap- 
idly; hence,  the  equations  are  not  valid  for  temperatures  above 
about  3000  degrees.  Pier’s  third  degree  equation  gives  a max- 
imum for  Op  at  T =»  4100,  a point  of  inflection  in  the  Gp  curve 
at  4767,  and  a minimum  value  of  Gp  at  T -5433.  The  second 
degree  equation  of  Lewis  and  Randall  gives  a maximum  value  of 
Gp  at  T ^ 3436. 

By  a slight  change  of  constants  it  is  possible  to 
move  the  maximum  to  a still  higher  temperature  and  still  preserve 
the  accuracy  of  the  formula  at  the  lower  temperatures.  The 


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62 


ec[uation  finally  developed  is 

X,"  7.41+3.5  9 - 0.48  9^  (A) 

r 

in  which,  as  usual,  9 '•T/1000.  The  value  of  9 for  a maximum 
is 

9*3.5/2^0.48  =3.648  or  T =-3648  degrees  absolute  F. 

The  graphical  representation  of  this  equation  is 
shown  by  the  full  line  curve  of  fig. 9 ,p. 61.The  dash  line  curve 
represents  the  equation  of  Lewis  and  Randall.  The  agreement 
between  the  proposed  equation  and  the  experimental  values  is 


shown  in  the  following  tables: 

Calculated 

Mean  specific  heat  Observed  Holborn  Equation  (A) 

per  lb.  between  & Austin 


20° - 200°C 
20° - 440°C 
20° - 630°G 
20° - 800°C 


0.2168  Regnault  0.2173  0.2180 
0.2306  H.  and  A.  0.2312  0.2310 
0.2423  " " ” 0.2410  0.2403 
0.2486  ” ” ” 0.2486  0.2479 


Joly  found  for  the  mean  specific  heat  between 
10° - 100°C  the  value  0.2120.  The  equation  gives  0.2116. 

Values  of  the  instantaneous  specific  heats  per 
pound  at  the  lower  temperatures: 

Temp.°G.  Regnault  Wiedemann  Langan  Holborn  Holborn  Equation 

&Austin  &Henning  (A) 


0 0.1870  0.1952  0.1980  0.2028  0.2009  0.2049 

100  0.2145  0.2169  0.2100  0.2161  0.2152  0.2169 

200  0.2396  0.2387  0.2220  0.2285  0.2289  0.2282 

400  0.2450  0.2502  0.2517  0.2488 

600  0.2690  0.2678  0.2706  0.2665 

800  0.2920  0.2815  0.2829  0.2814 


Pier  obtained  the  ne  an  specific  heat  at  constant 
volume  from  0°-  t°G  for  five  temperatures.  The  results  are 
given  in  the  following  table: 


64 


Higher  ^ ^ 

temperature  Observed 


1611  9.976  11.966 

1725  10.06  12.05 

1831  10.27  12.26 

1839  10.28  12.27 

2110  10.47  12.46 


Galoulated 

Pier’s  formula  Eq^uation(A) 


12.059 

12.168 

12.261 

12.274 

12.463 


12.059 

12.173 

12.266 

12.273 

12.456 


The  proposed  equation  gives  with  fair  accuracy  the 


specific  heats  at  the  lower  temperature  and  represents  Pier’s 
results  with  nearly  the  same  precision  as  Pier’s  formula.  Appar- 
ently the  equation  may  he  accepted  with  confidence  up  to  the  limjt 
T = 4500 Op. 


Water  Vapor;  It  is  well  shown  that  the  specific 
heat  of  water  vapor  near  the  saturation  limit  varies  with  the 
pressure  as  well  as  with  the  temperature.  However  at  these 
lower  temperatures  the  pressure  of  the  H^i^O  component  in  a gas 
mixture  is  usually  small,  and  some  low  constant  pressure  may  he 
assumed.  At  the  higher  temperatures  the  variation  of  the  spec- 
ific heat  with  the  prqssure  is  negligible. 

Data  for  the  specific  heat  at  the  lower  temperature 
are  furnished  hy  three  sets  of  experiments. 

1.  The  explosion  experiments  of  Langen,  from  which  is  deduced 
the  formula 

a;  X 7.9  t0.00215t 

r TT) 

for  the  mean  specific  heat  between  0°  and  t°C. 

2.  The  experiments  of  Holborn  and  Henning,  #iich  are  represented 
hy  a;  =^  8.43  - 0.3815 -lO’^  f 0.792 'id^t*" 

3.  The  experiments  of  Knoblauch  and  Jakob;  and  the  later  exper- 
iments of  Knoblauch  and  Mollier  extending  to  550^  C. 


i 


65 


The  discrepancy  hetv/een  the  experimental  results 
is  shown  by  the  curves  of  fig. (10),  page  66.  In  this  figure 
the  points  marked-fare  obtained  from  Goodenough’s  formula*  for 
specific  heat  of  superheated  steam  taking  a pressure  of  1 lb. 
per  sq.in.  The  straight  line  that  represents  Langen's  eq.uation 
runs  entirely  above  these  points,  and  the  Holborn  and  Henning 
curve  from  250  degrees  C.  runs  far  below  them.  That  Holborn 
and  Henning’s  results  are  low  due  to  systematic  errors  in  the 
experimental  method  has  been  asserted  by  Callendar  and  others; 
and  it  is  likewise  probable  that  Langen’s  straight  line  runs 
altogether  too  high  for  the  lov/er  range  of  temperature.  The 
results  obtained  from  the  experiments  of  Knoblauch  are  however 
worthy  of  entire  confidence. 

For  the  higher  temperatures  the  only  re]iable  evi- 
dence is  furnished  by  Pier’s  explosion  experiments.  Pier’s  form- 
ula for  mean  specific  heat  per  mol  from  0 to  t degrees  C is 

-3  -f  V 

8.065  4-0. 5-10  t 4-O.E'lO  t 

The  curve  that  represents  this  formula  (see  fig. 10)  is  almost 
coincident  with  the  Holborn  and  Henning  curve  for  temperature 
above  300  degrees  C.  At  very  high  temperatures  the  third  degree 
term  in  Pier’s  formula  assumes  importance  and  gives  rise  to  a 
rate  of  increase  of  with  t that  is  scarcely  credible.  Curves 
of  the  instantaneous  specific  heat  from  1000  degrees  to  2750 
degrees  G.  are  shown  in  fig.  11,  page  67.  It  will  be  seen  that 
Pier’s  curve  crosses  Langen’s  straight  line  at  t - 2050  and  then 
rises  rapidly. 

*Principles  of  Thermodynamics,p.l05.  3rd  ed. 


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68 


Lewis  and  Randall  have  proposed  the  formula: 

^f^=8.81  - 1*9'16^T -^2.22'IO^t'*' 
in  which  3!  denotes  absolute  temperature  on  the  G.  scale.  An- 
other formula  used  in  the  Babcock  & Wilcox  tests  on  heat  trans- 
mission is 

Yp  = 8.174 +0.Q576-10^t -f-O.BOSBt^lo’^ 
in  which  t denotes  ordinary  temperature  on  the  R scale.  In 
both  these  formulas  2Tp  denotes  the  instantaneous  specific  heat. 
The  Lewis  curve,  fig. 11,  agrees  with  Pier’s  curve  to  1750  degrees 
G.  and  then  runs  somewhat  lower  and  the  B.  & W.  curve  runs 
still  lower. 

An  exanoiination  of  the  original  data  of  Pier's  ex- 
periments discloses  the  fact  that  various  other  expressions  for 
^Tp  will  satisfy  the  requirements  of  the  esjperimental  data  quite 

as  well  as  Pier's  formula,  provided  no  attempt  is  made  to  sat- 
at 

isfy/the  same  time  Holborn  c:nd  Henning's  results  at  the  lovsrer 
temperatures.  We  attempt  therefore  to  construct  q second  de- 
gree equation  that  shall(l)  represent  fairly  well  the  Knoblauch 
values  at  low  temperatures;  and  (2)  give  values  of  mean  speci- 
fic heat  that  will  satisfy  Pier's  data  at  high  temperatures. 

The  following  equation  fulfills  these  conditiona: 

2Tp  > 7.03  tl.25  ©-f-0.2 

where  9 ’=•  T/1000,  and  T denotes  absolute  temperature  on  t he 
P.  scale. 

Reference  to  figs.  10  and  11  shows  that  the  curve 
of  the  proposed  equation  fits  the  Knoblauch  points  very  closely, 
and  that  in  the  high  temperature  range  it  shows  only  a moderate 


■w 


69, 

rate  of  increase  of  with  the  temperature.  The  equation  is  a 
fair  compromise  between  the  high  values  obtained  by  Langen  and 
the  obviously  low  values  of  Holborn  and  Henning. 

Methane : V/e  have  the  following  values  for  the 

mean  specific  heat  of  methane  per  mol  at  constant  pressure  for 
the  range  18  degrees  to  208  degrees  C. 

Regnault  9.507;  Wullner  9.106;  Lussana  9.483 
For  lower  temperatures  we  have  the  recent  values  of  Heuse: 

T^C  -80  -55  -30  5 15 

8.08  8.08  8.14  8.42  8.50 

For  the  instantaneous  specific  heat  we  shall  take  the  linear 
equation: 

= 6.03  t0.005T 


which  T is  in 

absolute 

degrees 

F. 

The 

agreement 

with  Heuse 's 

values  is 

as  follows 

TOF(abs) 

348 

392 

438 

501 

519 

Heuse 

8.08 

8.08 

8.14 

8.42 

8.50 

Calculated 

7.77 

7.99 

8.22 

8.53 

6^62 

For  the  range  18  degrees  to  208  degrees  0.  or 
from  524  degrees  to  866  degrees  F.  absolute,  the  equation  gives 
for  the  mean  specific  heat  9.504  which  agrees  with  Regnault  and 
Lussana. 

Data  for  the  higher  temperatures  are  entirely  lack- 
ing. The  equation  chosen  does  not  give  an  unreasonable  rise  in 
specific  heat  v/ith  temperature  and  when  used  in  the  equilibriu  m 
equation  for  the  reaction  C + 2H3^ ->•  GH^  gives  an  equilibrium  curve 


70 


that  agrees  as  well  as  could  he  expected  with  the  discordant 
data  available  for  the  equilibrium  at  high  temperatures.  We 
may,  therefore,  accept  the  specific  heat  equation  for  methane 
given  above  as  being  the  best  possible  in  the  light  of  the 
meager  experimental  data  available.” 

Experimental  data  on  the  specific  heat  of  gases 
other  than  those  treated  by  Professor  Goodenough  are  so  meager 
that  we  are  hardly  justified  in  establishing  a relation  between 
specific  heat  and  temperature.  Hov/ever,  for  the  gases  acety- 
lene, ethylene,  ethane  and  benzene  v/hich  appear  in  small  quan- 
tities in  ordinary  fuel  gases,  a large  error  in  the  specific 
heat  of  any  one  of  the  above  gases  introduces  only  a small  error 
in  the  totsl  specific  heat  of  the  gas  and  air  mixture  burned 
in  the  engine. 

Being  guided  by  the  fact  that  the  linear  relation 
for  the  specific  heat  of  methane  prodiJces  an  equilibrium  equa- 
tion that  satisfies  the  high  temperature  data  fairly  well  and 
also  by  the  fact  that  the  experimental  data  on  the  specific  heat 
of  benzene  which  is  a'VEtD.able  up  to  350  degrees  C.  is  also  sat- 
isfied by  a linear  relation  we  shall  assume  that  the  specific 
heat  equations  for  acetylene,  ethylene  and  ethane  are  also  lin- 
ear. 

Acetylene  ( 0 vH : For  acetylene  we  have  only  the 
two  points  by  Heuse  for  the  instantaneous  specific  heat  at 
constant  pressure  as  follows: 


j e(U 

. ■ : 

. «4  . u t r '•  > * 


.0  C'.  ''l;j  *o  a, 

V 

ot-tca*  ..  aiid*  , rcXoificit 


• y 

•i  : '-  r-j  ,-:v  t 


J :i!,  '. 


J 4-  sVDit*  r -ii  .’  r 


. rj  n . i.  j :-T#;  ^ « k 


'! 

t!  i' 


It  ^ • .•;  j'ili-.  ( 

«?*•  If'  <- !j  •’i  J > or»j)c  t I 

i 4.^.1 4>'-  ^ J’l'i  'u  j n>  ' I - 

'iC-"’'  01  J , ./•  •’-  ■ 'r  . 

-.i-  r ..  '.,  li  .’  w".  ai' •-'  « 


.i  ^ «-f  ^ c 

I I w < 4 4 P i ' 


T.  ,1  n ~:  'I  ' • ■ .“.  u'\T, 

.'  -i*. i t>  1 *;i4  t'-  ' u 

! 

'.  OW  J"'  . -■  I 

e t 

:.;r  • fc"'4  - oJt'ttoo  o ( 



; ■ A X !,  t?  • ' Tx^  iT  j. 


'.  9 C • 


*•  •:->  IX 


.'  -'I-.o  'ooy./'  ' •.•  X 'f- 


XI 


« X.  j WA«i 


T X..  ';'f.*.niij’i"o  fix  efiiJxJ 

■a'  '.c _ ’XI  < ’uW  L'flo  ii.' 

• wx*-»  *yTd  . *x *3la/ 


i.ol.f  - jIv-v  -J-'t.  : ••  i.  Vx;X  ..  - ■ u .\.  'i  oX  - 

- ^ j . -x;I*i'ri X Xiij i.»*  i . e<  *4iu  . o .-  Ot 

* i I 


vii.-.'  xl'y-:  rCt'-Jx  J o 91'Joc  xa«;r  r M f '.r.,,  >a  J .;'C^  x^olJ 

Oi  - . : u I '■..  - *■•  - f - o X&i)t  fc'flJ-  «>QX.'5 


.a  (■*,'I.-.  .1  -J  i..  ’lx  .'  C"l.x  <:  u '-y  t^i**  O'.t  i'jxX”  e-iisnsq  t ■ 'i 


IvjXi.;..  j M rx  ' .;jiu  :C*..'T  •:  -*•■  X' £ *'  %€  -*»j.1Sx 

- wtX  ' t'l.i  "'fiftfis/xi  x -.  3x'V;i  J , 'iCi  o 4^,vfj 

. iJC 

f 

:.V  1:7  7 S)’.  ■ .,?<>■£•.;  Of;  i A.  : ; ) lJ  It' 

■ ' J t»..  ^ ■ J"  C‘ .1  -*C»  , V 1 -^- 1* ’iX  . V ”i 0 £ ^ W». 


frt  / i r^j'^ 


t°G 

T^P.abs. 

+ 18 

524 

10.43 

-71 

364 

9.13 

The  straight  line  through  these  two  points  is  given 
by  the  follomdng  relation.  T is  in  degrees  Fahrenheit  abso- 
lute. ZTp  = 6.19  + 0.0081  T 

Ethylene  For  ethylene  we  have  the  identical 

values  of  Regnault  and  Luss&.na  of  11. 3E  as  the  mean  molecular 
specific  heat  at  constant  pressure  for  the  range  10  degrees  to 
£02  degrees  C.  and  the  following  instantaneous  values  by  Reuse: 


t degrees  C . 

T degrees  F.abs. 

y^observed. 

Yp  Gal 
culated 
from  eq 
below 

+ 18 

524 

10.22 

10.22 

- 36 

4E7 

9.17 

9.57 

- 68 

369 

8.79 

9.18 

~ 91 

256 

8.64 

8.41 

The  proposed  eq.uation  for  the  instantaneous  specific  heat  of 
ethylene  at  constant  pressure  is 
iTp  = 6.67  +0.0068T 

with  T in  degrees  Fahrenheit  absolute.  This  equation  gives 
a value  of  11.31  for  the  mean  specific  heat  over  the  range  10 
degrees  to  202  degrees  G.  as  compared  with  the  value  of  11.32 
given  by  Regnault  and  L^ssena.  The  comparison  with  Reuse ^s 
values  is  given  above. 

Ethane  :Ca.H<w  - For  ethane  we  again  h'^ve  only 
Reuse's  determinations  which  are  as  follows: 


t°0. 

T^F.abs . 

■yj,  fobs. ) 

( calc 

+ 15 

518 

12.40 

12.29 

~ 35 

428 

11.04 

11.27 

- 82 

344 

10.44 

10.32 

J 


u'  Hi 

XI': 


ox 


0 v: 


I 


\<  • ■ -■  •:  1 » * •*  - ' ^ 

-^^r,v->.  . ufiJ 

**'  • ' 


) - 


^^i  od^-  7 j:  V or-  r-.  ■"•*  - r^ 

'■•  ■ ■ * ■ •’ . :.  ' lu  r 

cr'  " • :.‘V"  ' ; 

. ) ;5/.  L'rv  t • ' ... ; 


r '■*■'9 


u 

» 


c . • ol.  • 


« . . r •».  _ 4. 


■*  •' *.  £ * ^ 7’'.  *'<*T 

f:ct  * •*  * V*'  r 

.r  c'!? 

. u^erT...  j- 


1 f. » p r 


e 

o 

• 

• _ :io  - 

«•  • 

' ' £’. 

. i 

-- 

^ m ^ w ^ 

W « w — ^ 

w;k  1C'.:  br^.ocL-L 

■*  ff  • * 

. - X ..  _ 0 r .*  ; :l  ; ..  4 a a . C i.> 

? * j » 

*■  • 

» ,-V  - ■ • ■ t • »> 

» 

f 

■ '-0 

.Jift-sit':  ■ ' :.'-‘:n6  ,:i  'i  :ij  In 

£ w 

# , 

» i L * V f 

•:  lU'  . 't  To  '^rXi  ^ 2 

. r 

Jm  <, 

■‘  ‘t  ■.- 

* t 

-;[j2  -.  J- 

r a ■ •' 

■,  :*r . '.  ■ 

Jl 

w' J!':  ;r‘f*r 

. /v»-Q  ;j|^'£r,y 

'1 . 

7 ;? 

'^V.0  OM' 

TC? 

; s.'-aWvfff 

4»  iS^M*  tm  •-  • •>  ••  v 

r:  c'lii 

jsr.f -+',-!»'rr^r 

o).'« 

f 

\ • i 

u c*)  .>  . 

'■O'-  y< 

■ '~n 

* Cl 

/t* 

* v«^ 

611-  • 

^ X. 

72 


The  equation  that  gives  the  above  calculated  values  is 


* 6.43  to. OUST 


Benzene  Vapor.  OgH^:  For  benzene  vapor  we  have 

data  over  a wider  range  and  can  therefore  place  more  confidence 
in  our  derived  relation.  The  experimental  data  are: 


Temp.rangg 


F ( ab  s . ) 


34- 115  653-699 

35- 180  555-816 

116-218  700-884 

350  1122 


observed. 

23. 337  (mean) 
25.913  " 

29.270  " 

38. 947 ( ins t.) 


Investigator  ^Calcu 
lated 

Wiedemann  23.34 

" 25.91 

Regnault  29.27 

Thiabaut  38.95 


The  proposed  equation  for  the  instantaneous  specific 


heat  of  benzene  vapor  at  constant  pressure  is 


?Tp'4.00  t0.0318T 
with  T in  degrees  Fahr.  absolute. 

A comparison  of  the  calculated  £jid  experimental  re- 
sults is  given  in  the  above  table. 

Graphical  comparisons  between  the  calculated  and 
experimental  values  of  specific  heat  for  methane,  acetylene, 
ethylene,  ethane,  and  benzene  vapor  are  given  in  the  figures 
12,13,14,15,  and  16,  on  pages  73,  74,  75,  76,  and  77  respectively. 

In  the  preceding  discussion  on  specific  heats  it  has 
been  assximed  that  the  specific  heats  of  gases  are  independent  of 
pressure.  While  this  assumption  is  not  correct  we  have  the  fol- 
lowing to  show  that  in  the  range  of  temperatures  existing  in  the 
gas  engine  the  effect  of  pressure  on  specific  heat  is  negligible. 

For  the  more  permanent  geses  we  have  as  an  example 
the  specific  heat  equation  of  Plank*  for  nitrogen  expressed  as  a 

*Physikalische  Zeit.  11-633  (1910) 


V % 


F 


i 


► 


VO 


1 9 C<i 


r 


•:.> . I 


«,  ( 

6 


1 .■ 


\ 


X.  . . ' * r J .1  J.U  J 

*,  » . • • ' ' ■i}' 

. : I, . ► : : . .,  •-  -U  ;■  'Hi:  i.  J”  , 

■•'D  ‘ 


•j 


J. 


J\  r- 


'■  : • ..•;<  .;  : ' i ; U:  t*Ci  4J  ‘ j - ’J..-;;^  rif'trj 

V ..  !'.  , C'  j':.:  ’j  i:  I . i. V " i:,  - . J VJ.it  • «'»v-  ; 'i. 

::  ^ :■  I- ; ^ v-  ;■  >;oI 

',  ; ;.  .:  !i  1 t s . .r.  : : •• f . ■:  . iv;n«  c 

■'  .;  r ; O'.*  ■:'••.  ; -i’-'-J'  -a '.  :.  O’J  Oi 

■";>  i.fi  - ■ u ^ •'•'•’  -'-'i  * i. ii'  * ’u:  t.- . -.  I.OC* 

'.  j;-;::)  " 


( 


i 


- 3nO 


BOO  isoa 


78 


function  of  pressure  and  temperature;  namely, 

Op  = 0.2246 -^0•000038T -f- 0.905-^ 

where 

0 calories  per  gram 

T Centigrade  degrees  absolute 

P pressure  in  kilograms  per  square  meter. 

Using  this  equation  Plank  gives  the  following  table  of  calcu- 
lated values  of  specific  heat  at  constant  pressure: 


Tsii^>«  Pressure  in  atmospheres 

Op  nn  0.0  op;  t o oo 


0.0 

o 

• 

to 

0.5 

1.0 

2.0 

3.0 

4.0 

-150 

0.2293 

0.2303 

0.2317 

0.2342 

0.2390 

0.2439 

0.2488 

-100 

0.2312 

0.2316 

0.2321 

0.2329 

0.2347 

0.2364 

0.2382 

-50 

0.2331 

0.2333 

0.2335 

0.2339 

0.2347 

0.2355 

0.2364 

0 

0.2350 

0.2351 

0.2352 

0.2354 

0.2359 

0.236S 

0.2368 

60 

0.2369 

0.2370 

0.2370 

0.2372 

0.2374 

0.2377 

0.2380 

100 

0.2388 

0.2388 

0.2389 

0.2390 

0.2391 

0.2393 

0.2395 

200 

0.2427 

0.2427 

0.2427 

0.2428 

0.2429 

0.2430 

0.2430 

Sat. 

temp . 

— 

-207 

-201 

-196 

-190 

-185 

-187 

Co  at 

sat. 

— 

0.2334 

0.2395 

0.2473 

0*2591 

0.2681 

0.2759 

temp. 

The  values  in  the  above  table  are  plotted  in  fig. 

1*7,  page  79t  At  50  degrees  G.  the  variation  in  specific  heat 
for  the  pressure  range  0 to  4 atmospheres  is  less  than  one  half 
percent.  As  the  temperature  increases  this  variation  decreases. 
In  no  case  will  the  pressure  in  a gas  engine  by  greater  than  4 
atmospheres  when  the  gas  temperature  is  below  200  degrees  C.  so 
that  the  error  in  neglecting  the  pressui’e  effect  on  the  specific 
heat  is  negligible.  We  may  safely  assume  that  this  statement 
holds  true  for  all  diatomic  gases. 

The  Lewis  and  RandEll  equation  for  the  specific  heat 
of  diatomic  gases  which  has  been  chosen  as  most  accurately  rep- 


\ 


' 


/ 


!:  3 * 


.1 

i L^r.  .1 


•HI'fT:  ; 

' 1 

;i  i 

.<  ’ 


80 


resenting  the  available  experimental  data  gives  values  fi)r  the 
specific  heat  that  are  about  three  percent  higher  than  those 
given  by  the  Plank  equation  for  zero  pressure.  Since  the  ef- 
fect of  pressure  on  specific  heat  is  less  than  the  discrepancy 
in  the  experimental  data  the  introduction  of  the  pressure  varia- 
ble is  not  justified. 

It  v/ill  be  noticed  from  the  graph  of  the  Plank  equa- 
tion that  as  the  gas  approaches  the  saturation  state  the  effect 
of  pressure  on  specific  heat  is  greatly  increased.  The  ques- 
tion naturally  arises  as  to  whether  or  not  the  effect  of  pressure 
on  specific  heat  is  appreciable  in  the  case  of  the  more  easily 
condensed  gases  such  as  CO 2.  c<nd  H2.O. 

We  have  Goodenough*s  equation  for  the  specific  heat 
of  superheated  steam  referred  to  before;  namely, 


06  i-  /3  T 


The  constants  are 


A m/7  (hi-O 

-7-  n t-r  f ^ 


Metric 
ec  « 0.3E0 
^ = 0.0002E68 
r ' 72!71 


English 
6C-  ^O.SEO 
(3  = 0.0001E6 
y - E3583 


Nx  4;  log  m =10.8E500;  log  Ea  =E. 53391;  A = 777, 


Using  the  metric  system  in  this  equation,  Goodenough 
gives  the  following  table  of  calculated  values  for  the  specific 
heat  of  superheated  sterdin  at  constant  pressure: 


u • 'a 


J:’  Z-.  i . ; •■ 


*->■>  - 


b-'J^Er 


f\'jf  ■'■  y *r  r 
7 _ %T » £>  l’« 
Cv  i 


cl'.xv  ■■y^s  j.‘.,  ..• 


.1  . 

»>  i.  Jr.  i'i : , f f.ru^ 


i 


u P';  :j  ■ L'  i"  'f 


'i  ».■<  * 


I . :;  7 f / ^ X JL 

p • fc*  '*•*«.  V A i t " I?  y \ 


Jat  J hrilJ 


• M W V* 


r.  UH  D {■"  J (;  J,  o 

0 e*. r.;  'r 


j i‘  U4  u » or  i .. 


A 


j 1 


Ui  J.-  > ‘ . C *?i/I.ua*'i  ' 

«o.;  lw  •'  \;I,(  vP  v<i.i  ACxv 

' C-‘Ii{,<»q;u  fi» 

i* 

r* 

':::  iDL>  .’jI.jOo  -! 

- ' - i i'JUOC'  ' 

• ‘ a .'c  .*!  i 


•;» . 


It'  ■ 

.v.-«  . c 


\r  <\ 


ti'iA  pr  .jQUPo  ciir 


t|* 


C*.'\  •%/  • 

''hj... 


oi'i^oX 


y'l. 


' ' 


1^1 

l•■r 


• vX 


■i.  J ; ^ 


4 1 « frii'  il(',  j ' X k 
■ !’*•''  j .{.■■*} '.’ .( X 


f 


■J  t.:c7x>i  , 


..  ro  J Ai  rJ’  -j  : J :^'£i  '.>3  ie ^.r 


’ 


,J  J -.IUM- 


81 


Temp 

• 

Pressure  Kg. 

per  sq. 

cm. 

°C. 

0 

2 

4 

6 

8 

150 

0.457 

0.490 

0.527 

200 

0.460 

0.479 

0.498 

0.522 

0.545 

250 

0.466 

0.477 

0.489 

0.S03 

0.517 

300 

0.473 

0.480 

0.487 

0.496 

0.504 

350 

0.480 

0.485 

0.490 

0.496 

0.502 

400 

0.489 

0.492 

0.496 

0.500 

0.504 

450 

0.498 

0.500 

0.503 

0.505 

0.508 

500 

0.508 

0.509 

0.511 

0.513 

0.515 

550 

0.518 

0.519 

0.520 

0.521 

0.523 

1 Kg 

per  sq. 

cm  -14.223 

lb.  per 

sq.  in. 

We  have  the  eq^uation  deduced  in  the  first  part  of 
this  discussion  for  the  specific  heat  of  water  vapor,  e function 
of  temperature  alone,  as  follows: 

* 7.03  +1.25  'id^T  +-0.2  -IO'^T"^ 

Changing  to  the  Centigrade  scale  we  have  for  the  specific  heat 
per  gram 

C^  - 0.3905  +0.125  *i6^T  +0.036  ■loS'^ 

From  this  equation  we  have  the  following  calculated  values: 

TempOC.  150  200  250  300  350  400  450  500 

C 0.450  0.457  0.465  0.473  0.482  0.491  0.500  0.509 

Plotting  the  above  values  we  have  the  curves  shown 

in  fig.  18,  page  82.  The  dash  line  is  the  curve  of  the  function 

of  temperature  only. 

Above  550  C.  it  can  be  seen  that  the  effect  of 
pressure  on  the  specific  heat  is  so  small  as  to  be  negligible. 

All  temperatures  in  a gas  engine  except  in  the  compression  phase 
are  greater  than  550  C.  so  that  the  pressure  does  not  effect  these 
specific  heat  values.  Assuming  that  the  initial  mixture  of 
gases  in  the  engine  cylinder  contained  lOfo  of  water  vapor,  vh  ich 
is  probably  above  the  maximum  in  any  actual  case,  the  partial 


f 


I 


' I," 


J ..  i 


■i. 


J 


\ 


I 


p f'  ( 


t 


I 

I 

I 


'J 


ti  c " ■ 


9-r  -'  ^ 


r- 


0? 


O r 


Jijr:;.'. 


1 


82 


pressure  of  the  water  at  the  heginning  of  compression  will  he 
about  1 Ih.  per  sq.  in.  and  at  a total  compression  pressure  of 
150  Ih . per  sq.  in.  fahs.)  the  partial  pressure  of  the  water 
vapor  will  he  only  15  Ih.  per  sq.  in.  The  temperature  at  the 
end  of  compression  will  he  in  the  neighborhood  of  400°C . The 
specific  heat  equation  chosen, v^hich  is  a function  of  temperature 
only, represents  values  of  the  specific  heat  at  very  low  pressure 
in  the  neighborhood  of  100  G.  and  at  pressures  of  from  10  to  20 
lb.  per  sq.  in.  in  the  neighborhood  of  400  C.,or  in  other  words, 
the  probable  range  of  pressures  of  the  Hjj.0  along  the  compression 
line  of  the  gas  engine  indicator  card.  The  equation  can  be  used 
in  gas  engine  calculations  with  the  assurance  that  the  error  in- 
troduced by  neglecting  the  pressure  effect  on  the  specific  heat 
will  be  within  the  experimental  error  of  the  determination  of 
the  specific  heat  at  any  given  pressure  and  temperature. 

Since  the  equation  previously  derived  for  the  speci- 
fic heat  of  carbon  dioxide  is  based  at  low  temperatures  on  exper- 
iments performed  at  atmospheric  pressure,  we  may  say  following 
the  same  reasoning  used  in  the  case  of  water  vapor,  that  the 
error  introduced  by  neglecting  the  effect  of  pressure  on  speci- 
fic heat  is  within  the  limits  of  the  experimental  errors. 


84. 

Specific  heat  of  amorphous  carbon:  The  experiment- 

al data  available  are  as  follows: 

1*  Regnault’s  value  of  0.2415  for  the  mean  specific  heat  of 

one  gram  of  wood  charcoal  for  the  range  18  degrees  to  98  degrees  G. 

2.  De  La  Rive  and  Marcet*s  value  of  0.1650  for  the  mean  over  the 
range  6 degrees  to  15  degrees  C . f or  wood  charcoal. 

3.  Bettendorff  and  Wxillner's  value  of  0.2040  for  the  range  24  de- 
grees to  68  degrees  G.  for  gas  rhetort  carbon. 

4.  Weber  gives  the  following  values  for  the  mean  specific  heat  of 
wood  charcoal: 

OO-24OG  0.1653  0°-  224OG  0.2385 

0°-  99OG  0.1935 

5.  Kunz  gives  the  following  values  for  the  instantaneous  spec- 
ific heat  of  beechwood  charcoal: 

435OG  0.243  1059°0  0.362 

561  0.290  1197  0.378 

728  0.328  1297  0.381 

925  0.358 

From  the  above  data  the  following  eq[uation  has  been 
deduced  for  the  instantaneous  molecular  specific  heat  of  amor- 
phous carbon.  T is  in  degress  Genti grade  absolute. 

Yp  =1.60 -^2.84«10~^T  - 0.57'10"^t'^ 

A comparison  between  the  experimental  values  and  the 
values  calculated  from  the  above  equation  is  given  in  the  follow- 
ing table: 


r. : 


^ I’O 


f f 


^ i:.  >‘i  'i 


a 

. Yf  ' . 

.L 

w->.  C * b 0'.»'i '.♦*.' 

at 

.:  '•  -icZ  / 

L' 

.V,  V' 

• ■ >’  £.V  c’ 

. :A  bE 

• 

:o  . . 'Tt  &f»<' 

•:r 

- f."  . 

- 

• :0  < ’Tf'/’ 

rx'?- 

. *.  . 

! 

XiOvO-i': 

• bVftoTJ 

t :•  ,:.•  0 ^ i'i  . . r , .•  .:r 

•■'.t 

':o1'  r.auj'  v A y 

- i!.i  ; 

Y-  X.  > 

c 

. » . - .1  r 

...  1.  ♦.  .1 

•'  > o ow^  'erto 
‘ Ti:  . 

' OXj  d 

- '-•  1 W L'  • .t 

vi,-  'letc.e 
: Ci  :i-00w 


-7*^*1 


^ Cf  \ •„.  , ' . , i 

:Pk  . 


, «-• ..» 


li  'io:  abj.rijv  . ,>  siOJ-i 

' T''  ^ ^Xi  lyi  ll 


..  j‘  . 


ij  r • 

■ .0 


.[‘ri 

^iSV 
,1  *-•> 


l«  •>  ■ r ! Xu  - 


•Xsi>  fsvo.i.j 


If  I 


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85. 


Temperature 

Molecular 

Investigator 

Centigrade 

Specific  Heat 

absolute. 

calc. 

Observed. 

291  - 371 

2.477 

(mean) 

2.898 

Regnault 

279  - 288 

2.359 

11 

1.980 

De  La  Rive  & Marcet 

297  - 341 

2.447 

n 

2.448 

Bettendorff  & Wullrer 

273  - 297 

2.363 

tl 

1.984 

Weber 

273  - 372 

2.456 

ti 

2.322 

II 

273  - 497 

2.607 

If 

2.862 

II 

708 

3.337 

( inst. ) 

2.916 

Zunz 

834 

3.573 

If 

3.468 

II 

1001 

3.869 

IT 

3.948 

II 

1198 

4.187 

II 

4.272 

II 

1332 

4.373 

II 

4.345 

n 

1470 

4.543 

II 

4.512 

II 

1570 

4.654 

n 

4.584 

II 

At  the  low  temperatures  the  equation  s trikes  about 
an  average  of  the  experimental  data  and  at  the  higher  tempera- 
tures agrees  very  well  with  the  Kunz  values  as  can  be  seen  in 
fig.  19,  page  86.  The  equation  gives  a maximum  of  5.138  at 
T 2490  degrees  C.fabs.)  or  4022  degrees  F. 

Changing  to  the  Fahrenheit  scale, the  equation  be- 
comes for  the  instantaneous  molecular  specific  heat  of  amorphous 
carbon 

=1.60 +1.58 'lO'^T  - 0.176 'lO'^T*" 


" i. : 


i , - 

j 

f ^ ■•  '■’ 

,JT 

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: . 

• 3 -■: 


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0( 


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87. 


References  on  Sipeoific  Heat 


1.  Bettendorff  and  Wullner, 

2.  Be  La  Rive  and  Marcet, 

3.  Groodenough, 

4.  Reuse, 

5.  Holborn  and  Austin, 

6.  Holborn  and  Henning, 

Y ^ n « H 

8 . J oly , 

9.  Enoblauch  and  Jakob, 

10 . Knoblauch  and  Mollier, 

11. Kunz, 

12 . Langen, 

13.  Lewis  and  Randall, 

14. Lussana, 

15.  Mallard  and  Le  Chatelier, 

16.  Pier , 

17. Regnault , 

18.  Swann, 

19. Thiabaut , 

20. Weber,  H.P. 

21.  Wiedemann, 

£2.  ” 

23.V/ullner , 


Pogg  Am  vl33,  p293.  1868. 

Landholt  and  Bomstein  Tables 

Principles  of  Thermodynamics,  3ed. 
Bul.#75,  Eng  Bxpt  Sta,Univ  of  111. 

Ann  derPhysik,  v59,  p86.  1919. 

Sitzungsber.  der  Kgl  Preuss  Akad. 
(1905)  pl75 

CO^.  Ann  der  Physik(4|  v23,  p809.  190*J 

H^.  ” ” ' 


(4)  v23,  P809.1907 
(4)  vl8,  P739.1905 


Phil  Trans  vl82,  p73.  1892. 

Mitteil  uber  Porschungsarbeit ,v35,pl09 

Zeit  der  Yer  Deutch  Ing.v55,p666.1911 

Ann  d.Phys.  (4)  vl4,  p309.  1904. 

Mitteil  uber  Porschungsarbeit , v8.1904 

Jour  Am  Ghem  Soc , v34.  1912. 

Huovo  Gimento  (3),  v36,p5,70,130.  189< 

Ann  des  Mines, v4,  p379.  1884. 

Zeit  Elektrochem,vl6,  p897.  1910. 

Mem.  de  1' Institute  de  France, v26, 
pl67.  1862. 

Proc  Royal  Soc  1900. 

Ann  der  Physik  (4),  v35,p347.  1911. 

Phil  Mag  (4) ,v49,ppl61,  276.  1875. 

GO^.  Pogg  Ann,vl57,pl.  1876. 

G^H^.  Ann  der  Physik, v2,pl95.  1877. 

Landholt-Eornstein  Tables# 


'( 

i • 


V 

t ’ ■ r 


? ' < 


it 


I 


88 


Appendix  B 
Heat  of  Combustion 

Hydro, gen.  Leaving  out  of  consideration  the 

work  of  investigators  previous  to  1848,  we  have  the  following 

brief  outline  of  the  methods  used  by  the  various  investigators 

since  that  date  for  the  determination  of  the  heating  value  of 

hydrogen,  and  also  the  results  obtained  by  each. 

Andrews,  in  1848,  first  used  the  bomb  calorimeter. 

Hydrogen  and  oxygen  collected  over  water  were  introduced  into 

the  bomb  in  the  theoretical  proportions  for  combustion.  The 

gas  mixture , under  a total  pressure  of  one  atmosphere,  was  ig- 

generated 

nited  by  an  electric  spark,  ^the  heat/being  absorbed  by  water 

surroiuidine:  the  bomb.  Andrev/s  found  as  the  average  of  four 

(0°C.  760mm.) 

experiments  the  higher  heating  value  of  one  standard  litre/of 
dry  hydrogen  at  20  degrees  G.  and  constant  volume  to  be3036 
calories.  (20  degree  calorie) 

Favre  and  Silbermann  in  1852  burned  hydrogen  v;ith 
oxygen  in  a closed  vessel  at  a constant  pressure  of  16  centimet- 
ers of  water  above  atmospheric.  This  burning  was  accomplished 
by  leading  two  metal  tubes  into  a small  metal  chamber, one  for 
hydrogen  and  one  for  oxygen.  By  regulating  the  flow  of  the 
^ases  a steady  flame  was  maintained.  The  heat  of  combustion 
was  absorbed  by  water  which  completely  surrounded  the  combustion 
chamber.  Since  the  flame  was  completely  enclosed  the  water 
formed  by  the  combustion  could  not  escape  and  so  was  condensed. 


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89. 

At  the  hegimiing  and  end  of  each  experiment  the  combustion 
chamber  was  weighed,  the  increase  in  weight  being  the  water 
formed.  Correction  v/as  made  for  the  non-condensed  water  va- 
por within  the  combustion  chamber  by  v/eighing  the  chamber  filled 

with  wet  and  then  with  dry  gases.  The  difference  is  the  v;eight 
non-condensed 

of/vapor.  This  weight  of  vapor  is  multiplied  by  the  latent 
heat  of  steam  at  18  degrees  C.  and  the  result  added  to  the  ex- 
perimental result.  As  the  average  of  6 experiments  the  higher 
heating  value  of  1 mol  of  hydrogen  at  18  degrees  G.  and  constant 
pressure  was  found  to  be  68,924  calories.  (20  degree  calories.) 

J.  Thomsen  in  1873  using  the  same  method  as  Favre 
and  Silbermann  with  minor  changes  in  the  apparatus  found  the 
higher  heating  value  of  one  mol  of  hydrogen  at  18  degrees  C. 
and  at  constant  pressure  to  be  68,357  calories  (20  degree  cal.) 
This  isthe  average  of  three  experiments. 

In  1877  Schuller  and  V/ertha  used  a Bunsen  ice  cal- 
orimeter wherein  the  heat  of  combustion  is  determined  by  measur- 
ing the  amount  of  ice  melted  at  0 degrees  C.  The  amount  of 
ice  melted  is  measured  by  the  contraction  in  volume  of  a mix- 
ture of  ice  and  water.  Volume  of  ice  melted  times  specific 
weight  times  latent  heat  of  fusion  of  ice  gives  the  heat  ab- 
sorbed by  the  ice.  The  closed  end  of  a test  tube  projects 
into  the  mixture  of  ice  and  water.  Inside  of  the  test  tube  is 
placed  a previously  weighed  glass  combustion  pipette.  This  com- 
bustion pipette  consists  simply  of  a small  glass  bulb  into  which 
are  led  two  glass  tubes,  one  for  hydrogen  and  one  for  oxygen. 


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90. 

Burning  occurs  at  constant  pressure  inside  this  hulh.  Since 
there  is  no  outlet  the  water  formed  condenses  and  remains  in 
the  hulh.  At  the  end  of  an  experiment  the  glass  pipette  is 
again  weighed  , the  increase  in  weight  being  the  water  formed. 

The  experiments  lasted  3 to  4 hours  so  that  the  small  amount  of 
vapor  left  uncondensed  in  the  pipette  introduced  a negligible 
error.  As  an  average  of  5 experiments  the  higher  heating  value 
of  1 mol  of  hydrogen  at  0 degrees  C.  and  at  constant  pressure  was 
found  to  be  68,250  calories  (mean  cal.  0)  degrees  - 100  degrees  G.) 

In  1881  Than  used  the  Bunsen  ice  calorimeter  with  a 
constant  volume  combustion  pipette.  In  order  to  get  an  appre- 
ciable volume  of  gases  in  the  pipette  the  calorimeter  was  quite 
large.  Hydrogen  and  oxygen  v/ere  introduced  into  the  pipette 
uiider  a total  pressure  of  one  atmosphere  (Barometer  760  mm.) 
and  exploded  by  an  electric  spark..  The  gases  before  combustion 
were  saturated  with  water  vapor  so  that  all  water  vapor  formed 
was  condensed.  The  average  result  obtained  from  5 experiments 
for  the  higher  heating  value  of  1 mol  of  hydrogen  burned  at 
0 degrees  G.  and  at  constant  volume  is  67,644  calories  (15  degree 
cal. ) 

Berthelot  in  1883  revived  Andrew’s  bomb  calorimeter 
and  much  improved  it.  The  bomb  v/as  first  filled  with  dry  hydro- 
gen under  a pressure  of  1 atmosphere.  V/et  oxygen  v/as  next  intro- 
duced into  the  bomb  from  a cylinder  of  compressed  oxygen  until 
the  total  gas  pressure  in  the  bomb  was  about  1.7  atmospheres. 

The  excess  of  oxygen  was  used  because  the  compressed  oxygen  con- 


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91 


tained  a small  percentage  of  nitrogen.  The  mixture  was  exjiloded 
by  an  electric  spark.  Since  all  the  water  vapor  formed  is  con- 
densed no  correction  is  needed.  Berthelot  gives  as  the  higher 
heating  value  of  1 mol  of  hydrogen  at  constant  volume  and  at 
10  degrees  G.  68,000  calories.  (lO  degree  cal.) 

In  1903  Mixter,  at  Yale  University,  made  further 
improvements  on  the  bomb  calorimeter.  Dry  hydrogen  was  first 
introduced  into  the  bomb  at  atmospheric  pressure  (bar.  14.743 
lb.  per  sq.  in.).  The  weight  of  hydrogen  present  was  calculat- 
ed from  its  knov/n  pressure,  temperature,  and  volume.  Dry  oxygen 
was  next  introduced  into  the  bomb  until  the  total  pressure  was 
in  excess  of  1-|-  atmospheres.  The  average  of  14  experiments  by 
Mixter  gives  for  the  higher  heating  value  of  1 mol  of  hydrogen 
at  18  degrees  C.  ruid  at  constant  volume .omitting  the  correction 
for  non-condensed  water  vapor  ich  will  be  considered  later, 
66,835  calories  (20  degrees  cal.) 

The  latest  work  done  on  the  heating  value  of  hydro- 
gen is  that  of  G.  Rwnelin  in  1907.  He  used  a bomb  calorimeter 
of  more  elaborate  designthan  those  previously  used.  Dry  hydros 
gen  and  oxygen  in  the  theoretical  proportions  for  combustion 
were  introduced  into  the  bomb  under  a total  pressure  of  1 atmos- 
phere (Bar.  14.32  lb.  per  sq.  in.)  As  an  average  of  six  ex- 
periments omitting  the  correction  for  non-condensed  vapor. left  in 
the  bomb,  the  observed  result  for  the  higher  heating  vslue  of  1 
mol  of  hydrogen  at  18  degrees  C.  rnd  at  constant  volume -was 
66,940  calories.  (20  degree  cal.) 


V. 


r>Pi  ^rn  .1  y,©*:  i*  j * iv  yi&':o:  ,.  JDeAi  .sv 

I ii  b/,y!r[->^  •'^Cfi«v  - _ V7  r.O  J.J  . Uiio  . x«vu  ...i.  , . 

V ■ 

IT  I ,^J.  i.J, ,.!*■“  **  ■*  0^0..  .' -' iv  '.  >J  \i.  • tA,.  Z.Z  4iv  V^Xl  jboci  _=Jj 

V ixii  or  i.'.ov  rn^jHitOi.  v*^  ii»  .u  *x  * ^iL  ic  X&;  _ X 1:  r;J^ 

S -J  ij  ♦.•»  CA  ^ * <«  .>^.  ..V  .MU  VJ'J  •»  , . ' • •.  ^/JO*^C*A  -»J- 
. rv* 

'i . w iJJ  X , '-j  X*  i.tji ^ X <i  ,w  , u.  iXl 

' J.3  V •!>»  3 1U  a ^tsx  « -CX.-^  'i»  V , T.'i;  ,1>.  j v ..aVw''  „ 1 

• •‘iflvJ*  cx‘'0<j.A^  v*x  •-*'i  w '^...  t.ii  . C .>  iVi.  2)8  SijXo*I  w 1 . 

■ ■ - »•  -•  V ii  K li  i*,»j  u Vu  »*  X .>  nil  ,qX  j T ■!*  iix  • \ « „ " » ^ 'i  * 

.u.  — o , ii-T  - -i.*’0fC3C'fl'f!  «o*ii  j:. 

■ 4 

-•  ^ ^ AM.'  ->  v%  ^ C \j , 1 L t I »)  * . X w 3C6  i}  Ai 

. ‘rf . ■ 


V/**:  x ma  A-  1‘A:  . . -Xv2:. M*  .4  '';;I  ..l  '-j^coce  i\i 

.'>.  . ".xt.  XV.  luM  .:.  .w  u uu .'  -\  1:  Si1  •ii.^^iX  i xiu  n/Jt  ,t.  '.ly  ‘lOJXl.' 

j 

-i  - XiV/2.  4>i4  4 C^**x  V ,»--wXc.  / g-.i'.'w'.  >c  .".i  •..  j’i>- 

• '-  - ^.  ■ ■ X k.  i.j  Cj_0  V n V.  .*.  » .,  * , .0  vj,4*  V '1  ■.,  J . , c “•  .1  s'ii  Tt.  . 

■■  . , <'di  i i.  •..  . / 1 ■■.•  X-l^  C ^ ^ ■ 

i ~ y:'  ^ •■’  ’■•*■*  Xi.'  .'  ■•  »x  v' 81*.  ?13'>  ^ 'tw  n - W ’JIa  A 

V ...  . . ;.  : 'I'nr.  .•;  '.X  r.i  r,U£oi  . ti, /:  * 

' .’  - .:  . • - .-  iClax;  I;*.-',  "'nj  * ’'‘•or  jm  Xo 


i-.w  lo  t-u'vi:. . i *rC'x  *>;iv.i  w‘"'0c4,c\' : r,  ,ci.J‘yiOi.  ..  r_'{-  f.i.  r ' fiffc  ' 

V.  r . .!■'  V. 'i  ^ X ^ .Z  o^f  .*■  tn.l:  „8'ti»*.. 

-■  Im  'xO  o"3*idv  • : r.  (.  "X  . ■*  tMf  .^1  . vf  .*c^5) 


-0  ■*  .jasi'a Xnc c - t).' Av  c.a'.'ro-.  -.:t  ' 

'¥ 

•.  X T - i:..  J : ?.{  ’tf„  T"'..  v^X.7s?T  ■«'ft 


A.  ; / n U-  v;  ; . ■ 


.V  sao'iaoii  3i  ‘j.  t T'^'h 

■ ' ^.'  ■ 

( . Li  c or-J’  ■'. a ^ ?•  T3  : . , D Cl'  'i . ^0  ' 


92 


In  order  to  put  the  atove  values  on  a coniparative 
basis  we  shall  ah^mge  them  all  from  the  French  system  to  the 
English  system  of  units  and  use  the  mean  B.T.U.  in  the  range 
32  - 212  degrees  F.  Also  the  lower  heating  value  will  b e cal- 
culated in  each  c§.se. 

To  convert  to  the  mean  B.T.U.  we  have  from  Callendarfe 
eq^uation  for  the  specific  heat  of  water  taking  the  mean  B.T.U# 

(32  to  212  degrees  F.)  as  unity  the  following  correction  factors; 


Temperature 

Correction  factor 

log  correction 

Degrees  C. 

Degrees  F. 

tor 

10 

50 

1.00150 

0.0006499 

15 

59 

0.99962 

1.9998352 

20 

68 

0.99842 

1.9993100 

V/e  shall  use  the  following  notation; 


Hv 

= lower  heating 

value 

per 

mol 

at 

constant 

volume , 

n 

IT 

IT 

IT 

If 

XX 

pressure , 

h; 

= higher 

XI 

IT 

II 

XX 

Tf 

XX 

Volume , 

h; 

XX 

It 

If 

TT 

IT 

XX 

pressure . 

The  experimental  methods  outlined  above  fall  into 
the  tv;o  general  classes,  i.e.,  burning  at  constant  volume  and  at 
constant  pressure. 

To  deduce  the  lov/er  heating  value  from  experimental 
results  given  we  shall  use  the  folloif/ing  method;  As  the  hot 
gases  are  cooling  after  combustion  no  condensation  of  tlie  water 
vapor  occurs  ULitil  the  temperature  of  the  gases  falls  below  the 
saturation  temperature  corresponding  to  the  partial  pressure  of 
the  v/ater  vapor  constituent.  Condensation  then  occurs  between 
the  saturation  tempera,ture  and  the  final  temperature  of  the 


M 

I 


93. 

gases.  If  the  initial  gases  were  saturated  with  water  vapor, 
all  the  water  formed  by  the  combustion  will  condense  if  the 
products  of  combustion  are  brot  back  to  the  initial  temperature 
since  the  amount  of  water  vapor  required  to  saturate  a given 
volume  of  any  gas  is  dependent  on  the  temperature  only.  If 
the  initial  gases  are  dry,  however,  that  portion  of  the  water 
vapor  formed  v/hich  is  required  to  saturate  the  given  volume  of 
gas  at  the  final  temperature  will  remain  as  vapor.  To  get  the 
higher  heating  value  when  the  initial  gases  are  dry  the  latent 
heat  of  the  non-condensed  vapor  must  be  added  to  the  observed 
results.  By  definition  the  lower  heating  value  of  a gas  is 
the  amount  of  heatgiven  out  d'oring  the  combustion  of  the  gas 
considering  the  water  formed  to  remain  in  the  vapor  state  when 
the  products  of  combustion  are  cooled  down  to  the  initial  tem- 
perature. . The  difference,  then,  betv/een  the  higher  heating 
value  and  the  lower  heating  value  is  the  difference  between  the 
amouat  of  heat  given  out  by  the  hot  gases  in  cooling  in  the  first 
place  with  condensation  and  in  the  second  place  cooling  without 
condensation. 

The  heat  given  up  with  condensation  at  constant  vol- 
ume in  cooling  from  the  satiiration  temperature  to  the  final  tem- 
perature is 

,Q^  -=  ■‘i,"  tx  f*’) 

where 

u'' -^thermal  energy  of  1 lb.  of  steam  at  the  saturation 
point  in  the  cooling  of  the  products  of  combustion. 


f ■ SI 


. tj  * ‘ • > v«  X , ' l ‘j  * V*  s ^■ 

^ j U'l 


J.  . M 


■>x 


:x 


;-i-y:i.r 


f'l 


* 1 k ♦ I . 


i : f 


^ ; s/i  ’ 'j  0 i.  My*i , ^4,j.  • 

■;>.  .V(;.  'C  UJ^P^C/.  , 

' * / *A  -- 0 T;  .»^  ’**^1  J Ci*J  V- 

; . .a;. ’'itiv  t.i  L.J.'  : c •^rorlov 


V-*  ^ w 4-  - 1 ,'*■  U*J 

•'  k ff. 


V 


I - 7 .. 

f?'- ; 

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f 

,'V‘  L ,.  , '.1  ■ 

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• 

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-X>i  «#•'  , • 

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4 -i 

I.i  : iy  . - . VI 

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;*•  J.--ru’:  o.-’j  c.^'X* 

. oC 

. i.Ct. 

..  X 

4,  X ..  J ' .'\u  i . J »'  f^V 

r.'.'v  ?„n.t-xvr-ri  •xtPi.aXr 

■j:.\'  ..I  . 4*.'i.ii 

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X 

.;:.  u - V '. 

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ij 

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C !:  Jf  i i. 

'At. ' w f!  ,'i  ■ -^  .1 

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:xi>X  ' . 

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ii" 

r : : 1:  t 

. 

■ ',  '’c  .j:-rj7ci  "'''o 

. V t x^  '.X'ier:  pfiv ; 

- • X A.U  I'.'X  ., 

:,  c>..-  rr-Yox  x. 

sX'- 

*■  ')'x  ';E  ' •■  P-Jx.'-Xt 

0 .'>r  t'.iA 

X X 

* ‘ * 

J-  ■'•c.'r* 

. ,.’XxJ  ^-ieq 

' iX />  'w  x<I 

• 

V i;  rK-J 

t 

4 C 

,-i.  '■  ,'  t ;flU  'X«  . X •->  ..,  jutv.  J ' ..  V 

u J -I .. 


■;  (>  ./  li  OiijUt 


J ’-jrf  . ^4  ^ ' j f w -J  W«4V  J '%’  kJ* 

i ■ • >' 

jT’  ilu (.;  t» .'J-;  i i-J.U  5L.-,C>i 


J j/'.' {!:.!.  7 -aijXocc  i;-‘f  -■:• 

»•. 


,/ 


*'  j.  ■•  U ; \i  oZlC'O  w *i 


.a'sJ'il'n  . f.-s)  j , Vx  -j  t*:'  'O  '' 


~ i-  L '\J  X 


4*iiw  v>u 


A*£^,rrj  /I. 


;.  ?■;,  j r.j'i”.  '.ixJ.  (iQti  ni  oiax. 
.(  • ' _..  EX  ^^'XX/^^' i“x~  I 

• ,'  ■ , I,- 

( X L )~  .U 


.'■'.  ''  V J 


iX '.  Sij  O 'xXj  J IX  £? ..  J 


. L £ to  • i^'xrr-%^:  .X'X;  i; 


• L.I-  ',.w  c*.  7':..;  ,.;Ic5-  ’;u  ?„riXj.tOu'  ^'^r 

' ■ r 


■r  'r:t  j-{:loq 
■:,4 


sr-  ■,.'sr''-.t:rmr-r.gi 


SET 


94t 

11^=  thermal  energy  of  1 lb.  of  liquid  water  at  the  final 
temp  erature . 

X - fraction  of  water  formed  remaining  non-condensed. 

^ - intern?.!  latent  hert  of  1 Ih.  of  steam  at  final  tem- 
perature . 

The  heat  given  up  by  the  v/ater  vapor  in  cooling  at 
constant  volume  from  the  saturation  temperature  to  the  final 
tenii-)erature  considered  as  a non-condensable  gas  is 

1^  9.  ~ j 

where  r=.  instantaneous  specific  heat  of  1 mol  of  water  vapor. 
Hepresenting  specific  heat  by  the  following  fuiction 

= a +bT  +fT’" 

we  have  upon  substitution  i nd  integration 

= a(T.-  T^)  +ib(T,’'- 

2;  - H ^ 

+tb(T,''-Tp+if(T,’-T’^ 

The  temperature  at  v/hich  saturation  first  occurs 
in  the  cooling  process  at  constant  volume  is  determined  b;v  the 
following.  Knov/ing  the  pressure,  temperature,  ‘„nd  composi- 
tion of  the  initial  gases  the  total  volume  CcJQ  be  determined. 
This  is  also  the  volume  of  the  resulting  waiter  vapor  from  which 
the  volume  per  lb.  is  easily  determined.  The  saturation  tem- 
perature corresponding  to  this  specific  volume  can  be  formed  in 
the  steam  tables  ana  is  the  temperature  desired. 

The  heat  given  up  with  condensation  at  constant 
pressure  by  the  gases  in  cooling  from  the  temperature  at  which 


r 


fjrr- 


.1 


> 


j: 


. T X '•.•/.  jrii  X-'.  ’.r 

• *'  *^X  * * 'J  2 \l  U 

I '^<54  'itC'.  .U/ I v , . ■/*  I *’  ^ 

X to  j::  X'tvJi'l  C *rt- ,,,j^  ; ^ ^k. 

' . •.-v;J’.:*'xe''j' 


•V  -.iv^  , 

. ' '.  >.1/  ft/  . 


iX  ■<  • 7 V e(‘ 


‘ 0 ••>  . 

■:.'»■  Jj 

T v arft-o  1 

t 

\iOa 

dtir/  ! 

V 

./.  ■ 

> 

\ • p. 

/ .>  .>W  «,  'V  ^ 


. ’ I 

. ^ J uX  iX  t'G-j'i-  *i- .'  Vj  ■,  u’;i-.;'j  >,."X  - ^ fk’:a._ 

i.iv.Ciiv^X  Ri'.J  jJ  J v^x.-.:  / •! 


iw  .r.c.«'‘A  . 

•« 

.-.  '•  • • 4»v; 

#ti4  Off 


j 


"'  ^ c.  ; 


/ 


I f _ 


( ■ “ (■- 


i«  •• 

- 1 (. ..  i/jrj  J 

A _ 

- .-)J  x r ( 

\ ~ ' )li  - 

. 

- 1l 

/ ’ 

,'fj  ••  - 

• 

f 

'€ 


jx 


b" 


.. ^uvjc  J ,‘ix X ii-  'ix/j  - 

i'J*  .,u  X,  -fV-i'X  V.  1 -LI..M  /•  / ......('li  J".  . i-L-Olv  y:l\X  f.(0:  « li/  iU 

"■  J V-  i • 1.  - ii » • , . I "1  I.'  • , . L 

.'.,  ■/»  >»  V-  ^•■1  *J 


'xn 


I'-t  V JxiiXJ 


1 V 

0-v  . 

^5J-XWO^X: 

, 

'.vuXioi 

W o/  ai>*. 

kW  ^ 

. *• 

X V X >tx 

:rc 

% .• 

Xvc 

r;ao*!  [jr 

» W vw 

y ' 

ru- 

'CV  4::1X 

O 

X !i 

:,n 

x;Xi 

' ‘ t 

ii/:  J 

X . ,C  Ufi 
/ 

> ^ ^ • *• 

ov 

oiiX 

'•sc.  : iiX 

t 

4J 

OJ 

X ur.ci  %,  I'j 

. :0  0 e*; 

i;/  • 

XLCr 

S.  . . . u '.• 


/ r'.;  « . C 'XJ.  V. 


i / 


- t > ■ <f  >- 

I .v<w.w  . -. i.{/ '•  : 


- t»i  ;eX  r.  i/e  . T 

i ' J ■ixs-vi:A  f.n'T 


V » i'  • • J .•(.*• 


fTto*!!  ^.nXXOcp  .'X  £Cl’.,2,  tr.s 


*-r  ymcm— ai 


95. 

saturation  is  first  reached  to  the  final  temperature  is 

- C “(ii-^^cr  ) 

where 

l'/  - thermal  head  of  1 Ih . of  steam  at  temperrture  in  cool- 
ing at  which  srturation  is  first  reached 
i^  = thermal  heed  of  1 Ih.  of  liquid  w?ater  at  final  tem- 
perature . 

r latent  heat  of  vaporization  of  1 Ih . of  steam  at  final 
temperature . 

X - fraction  of  water  formed  remaining  non-condensed. 

The  heat  given  up  by  the  weter  vapor  in  cooling  at 
constent  Jjpessure  from  the  saturation  temperature  to  the  final 
temperature  is 

=/V 

Xp  - instantaneous  specific  heat  at  constant  pressure. 

Tp  * a'-H  hT  +fT'*" 

,Q^  - a'(T, 

h;  - Hp.^;  -(i;+sr  )J-[a'(T, + + 

Knowing  the  pressure  at  which  combustion  occurs 
and  the  composition  of  the  products,  the  partial  pressure  of 
the  water  vapor  in  the  products  can  be  determined.  The  sat- 
uration temperature  corresponding  to  this  pressure  c?-n  be  found, 
from  the  steam  tables  and  is  the  saturation  temperature  desired. 

The  calculations  of  the  lower  heating  values  for 
the  constant  volume  c ses  are  'S  follows: 


d"' iiJ 


I 


* -jI.’ c i.  .:  i i.. 


i-  0 * V *,  1 * *T*  .'v  *T  J o 

^'0-:iL-  . • .' 

~ ' r . . , r . . . 

- - • > -•.  .1X^0.  , 

i.  I I ^ L’  V :• 

• •■  O-  . X'-  ’l  -' 

i ...  ji  I 'i\. 

■it.. I 4,  &C^4  wj  ,J 


-'  '• 

• Li  X 

' .■ 

'.  X;-: 

w • 

xXj  -a/X 

-Xo 

£ 't 

. ."  £10 

t 

jtr  X.S 

"1 


X tc  a'  ■ J”  r-v  V 


. . :..7  .. 

■..'fi  ur»3v(‘-  £ «»  *1 


: ' • ^ yi.r^c ; , r:o.:..o  -l  i - x 

;:i  ' aov  . ^>ri  srfT 

,:r  I.j ,3'ia J'c.r,  uiw  'uil  oiuar.»'iq 

■■  ' * • “>s 

j t W . i *1'^' V 


w ■;‘<i  ."k't  J" vi. . X 0 0 . ui\  v»;*  Vsf.  »Ti..  u -L^n.i: 

■ - 

),)  ^ (/•-  Z)"^ 


(-'■.-'  ■■  ■■  ' 


. •* 
, 3 


V - , ;.■:  .(■>  •.)(<„•  ^(  '•-,’)  .,  -■(  v::  -'l)-  ;...  t'  - ;h  ^ 

t.-  :..,o>-  ; ^ »)'o  uii  .;,nl 

'io  B.;‘xq  b'iw*'  • tr- r , ■C'jiibZ'z  }o  1 >Mu  i .c -^'.};>.o  ijrtJ*  bfiil 

* 'V> 

-V  ,!a  etIT  . 3tnjLr,Ti;  Xo/j  .,  ; ix.  t-  'juC«:i.ix.’i:  '..i-,  ;i.j.  'lOcAV  'xeJ'xiw  t«xU 

t l,  ; -;.,  '. 


Ci'X  a 0 0 fuf., 


,'  #r  f*  ,-  < 

w O t . ^ V.  .'. 


iiii/ji'C-  L j’i  . C 0 w* i'.  yiJ'L.i'SiJ 


1,'fuj  'cj  0/;/  a;  ,6A;j’‘e'!x.Vv  p-u. la  ^ci’t 

yeifX  •“  laH'oX  onV  ";.c  ■ I:;v  l v 

: :'OI,COa  i.i  ..  l>-  ;■  . 7 Jil  v:  ’ I'.'O  e-iX 


^6 


Andrews.  Since  the  initia.1  gb\ses  i_re  imder  atmospheric  pres- 
sure and  saturated  v;ith  water  V£  or  the  reaction  is 

H^O  -7(l-y)  H^O 

From  Groodenough’ s steam  tables  the  saturation  pres- 
sure at  68  degrees  F.  is  0.339  lb.  per  sq.  in. 


^ 0.339 
1.5  y 14.7 

Solution  gives  y 0.0354  mols. 

Volume  of  1.5354  mols  at  68  degrees  F. 

= 1.5354  -^380.6x111  = 596.1  cu.  ft. 

380.6  cu.  ft. = volume  of  1 mol  at  62  degrees  F.  and 

per  sq. 


14.7  lb. 
inch . ab  s . 


Volume  of  1 lb.  of  Hj_0  in  finul  mixture 


^ 596.1 
18a1.0354 


31.715  cu.  ft. 


The  saturation  temperature  corresponding  to  this  volume  is  E03°F. 
u,"  = 1076.2  u^  = 36.1  x=0 

,0,'  = 18(1076.2-36.1)  =18 . 722  B .T.U. 

^Q^-^^'^dt  = 5. 04(663-528 ( 6 63 *"-528^ 10~^^ 
(663^-528^) 

=791 

= 18722  - 791  =17,931 

Ey'  = 22.412 ^036 >«1. 8 ^0.99842  =122,000  B.T.U.  per  mol. 

=^  122,000  - 17931  = 104,069 
Hp  = = Hv  -f  T 

Hp  =■  104,069  -#-528  -104,597  B.T.U.  per  mol  at  68  degrees  F. 
Than:  Method  same  as  for  Andrew's  case. 


r 


T 


t!‘.  ® ‘'S  .fi  r ..x . : V L .•  . • 

^ vr  T "jt  * Kt ; * oi  -itaJ"  n f>ffj  n > 

,^-lJ  , U 'J£ 

oiJTtycrtd’esIflJ’  “£./■’ 

. t . j * 3|  .1£  ix  . j„3.«  d /6 



• . I • ■ ( 

i! 


i V <k  r.  V i '>  •▼«►>•  •< 


N/i  /I  i ^ i 


<<•»  ’ 


4.  q 


‘ r •"  ' . ; •• 

^3.  i.  M * tk  J,  V < 

I , ^ ;n  rrf  r. 

. > - 

r > ^ ■ r '•  - • I ' 

r ~ 

<•  *n  •■  - •■  »»  f ^ ^ 1 " 

• ...  , . 


A -k.  ^ W «^Uk  4 ik  V • 

— » -f  ~ 

• . ,if 

-»  * fy 

• ■ ' . W*  , 

. '“  Own.) 

.c:v  r. 


JL'«*  V4^ 


M ■ 


• .Vi'lJ  V.  • W • ♦ • W ' O' 


- J w f X 


«A*%>  « • wu 


W w ■ w <#*  X 


*.  * .^  .*.  ' • * 


u-  X 4 


«x  •»•  wwOjVwu 


. I 

OO  %»  >a4. 


' a *-jTf  * M j 


• , - f' 

. t (.  . - 


nt 


97 


Initial  temper'  tiire  =-32  degrees  5’. 

3c turation  pressure  at  3£  degrees  F,  - 0.0887 
y » 0.009 

Volume  of  1 lb.  water  vapor  in  products^ ^ 2^0  cu.ft. 

18  1.009 

Saturation  temperature  *=206.3  degrees  F. 
u/  =-1077.2  uj  = 0 X = 0 

18x1077.2=19,389 

,Q,y,  =1016 

eJ  =67644  xi.8>  0.99962  =121,713 
= 19389  - 1016  =18,373 
-=  121,713  - 18,373  =103,340  B.T.U.  per  mol. 

=103340  +492  = 103,832  B .T.U.  per  mol  at  32  degrees  F. 

To  change  this  value  to  E at  62  decrees  F,  we  have 


~ 6.5  '^0.5  .10'^  T 

= 3.25  t 0.2778  >10~^T 
3u/77  -TrTTTUTTfWTTaW^ 

frn,.  = 7.03  -M.25 -ld^T+0.2 -IO'^T"^ 

P/T  = 2.72-  0.4722*10-^']?  - 0.2'i'd^T’' 

[2.72(522-492)-'0.2361*10'^(522'’-492^) 

- £l2.10'^(  522'’-  492*^  )]=  -f-  74 

3 J 0> 

Hp  — 103,906  B.T.U.  per  mol. 

Berthelot: 


Initi'l  partial  pressure  of  dry  hydrogen  -=14.7  Ih .per.scL.in. 
Tot:  l pressure  of  initial  ga.ses  about  1.7  atmospheres. 


■T' 

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98. 

Initial  fnd  final  temperatures  = 50  degrees  P. 

Initial  gases  saturated  with  H^O 

Since  the  p^.rti^il  pressure  of  the  dry  hydrogen  in 
the  initi-^1  gases  is  1 atmosphere , the  partial  pressure  of  the 
water  vapor  formed  in  the  products  will  he  1 atmosphere,  at 
50  degrees  P.  when  considered  non-condensed. 

Pirtial  pressure  of  v/ater  vapor  used  to  saturate  initial 
gases  at  50  degrees  P.  -0.18  Ih.  per  sq.  in. 

Therefore:  Totfl  partial  pressure  of  H2.0  in  products  at  50  de- 
grees P.  - 14.70  i-0.18  = 14.88  Ih.  per  sq.  in. 

Volume  of  1 lb.  of  water  vapor  in  products  of  combustion 

18  5££  14.68  "*20.408  cu.  ft.  per  lb. 

Saturation  temperature  = ££7  degrees  P. 
u/'  =1083.0  u^  =18.1  x=0 

= 18(1083.0-18.1)  = 19,168  B^T.U.  per  mol. 

= 1037  B.T.U.  per  mol. 
hJ  = 19168  - 1037  18,131 

= 68,000  ^ 1.8  1.00150  »1££, 583  B.T.U.  per  mol. 

Ey  = 1££,583  - 18131  =104,45£ 

Hp  = 10445£  +510  =104, 96£  B.T.U.  per  mol  at  50  degrees  P. 
Mixter: 

Initi  1 gases  dry  at  64  degrees  P. 

Initial  pressure  of  hydiu gen  =•14. 743  lb.  per  sq.  in. 

Pin^l  pressure  of  water  vapor  formed  considered  non-condensed 
is  therefore  14.743  lb.  per  sq.  in. 


99 


Volume  of  1 lb.  of  v/rter  va'oor  in  products  of  combustion 

^ 580.6  ^524  ^ 14.700  , ^ .. 

" 18  522  14.743  “21.164  cu.  ft. 

Saturation  temper..-oture  =-225  degrees  S’,  (from  steam  tables) 


u^  =1062.5 


u^  s^32.1 


f>  = 998.4 

Saturation  pressure  at  64  degrees  P =•  0.295  lb.  per  sq.  in. 

Therefore:  = 0.0200 

14.743 

=18|l082*5  - (32.1-^0.02*998.^  ^ 18547 
/Qj.  = 945 

-Hy  =18547  - 945  =17.602  B.T.U. 

(observed)  = 66, 835Jil. 8^0. 99642  -120,112  B.T.U.  per  mol. 


=120, 112  - 17,602  = 102,510 

Hp=102, 510+524  =103034  3 .T .U.  per  mol.  at  62  degrees  F. 
Bumelin: 

Dry  hydrogen  and  oxygen 

Tot;:l  pressure  *1  -atmosphere  =14.32  lb.  per  sq[.  in. 

Initi  1 temperature  =64  degrees  F. 

P- rtial  pressure  of  water  vapor  as  products  non-condensed 

= 2/3x14.32  =9.55  lb.  per  sq^.  in. 

Volume  of  1 lb . w-.  ter  vapor  ^ x|24  =52.673 

18  ^22  9.55 

Saturation  temperature  -202  degrees  F. 

Saturation  pressure  at  64  degrees  F.  = 0.295  lb.  per  sq..  in. 

X = 0.295/9.55  = 0.0323 

u/':^  1075. 9 u/  =32.1  ^=998.4 

=18|l075.9  - (32.1+0.0323*998.^  - 18 , 209  B . T .U.  per  mol. 
= 807 


■X  o 


t.' 


ifi/l'.v  JOwTt* 


O' 


Y 


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> a 


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100 


= 18209  - 807  =17,400 
(Observed)  - 66, 940 >1.8>0. 99842  - 120,300 
^ 120,300  ” 17,400  = 102,900 

Hp  = 102,900  f524  =103,424  B.T.U.  per  mol  at  64  degrees  F. 

The  calculations  for  the  lov;er  heating  values  in 
the  constant  pressure  cases  are  as  follov/s: 

Favre  and  Silhermann. 

Temperature  of  combustion  » 64  degrees  F. 

Pressure  of  combustion  =16  cm.  of  water  above  atmospher^ic 

= 14.70 -hO.23  - 14.93  lb.  per  sq.  in. 

Since  HjO  is  the  only  product  of  combustion,  we  have 

Saturation  temperature  at  14.93  lb.  per  sq.  in.  = 213 ®F. 

i/  =1152.1  =32.1  X = 0 

= 18(1152.1  - 32.1)  ^20,160  B.T.U.  per  mol 

+1.25  -lO^T  +0.20  -lO'^T*" 

= 7.03  (673  - 524)  f il.25-10^(  673"’  - 524^) 

1o'^(  673^  - 524^)  =1169 

3 

Hp  - Hp  =20,160  - 1169  = 13,991 

Hp(observed)  = 68,924  ^1.8  XQ. 99842  = 123,866  B .T  .U./mol64°F . 

Hp  = - /r/f  / ^ -$0  rry  .T.  per  mo / aT 

'Shomsen: 

Temperature  of  combustion  =64  degrees  F, 

Pressure  of  combustion  =14.70  lb.  per  sq.  in. 

Since  H^O  alone  is  the  product  of  combustion,  the  pressure 
of  in  the  products  = 14. 70  lb.  per  sq.  in. 

Saturation  temper^iture  -=212  degrees  F. 


ii 


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101. 

i/'=  1151.7  = 32.1  X i 0 

,Q/  - 18(1151.7  - 32.1)  = 20,153 

= 7.03  (672-524)-^-sl.25'10“^  (672^-524’") -f%2(672^-524^]*/^>'^ 

3 

^ 1162 

Hp  -Hp  = 20,153  - 1162  ’=18,991 
Hp  (observed)  - 68,357 XL. 8^0. 99842  =122,849 

Hp  =122,849  - 18,991  =103,858  B.T.U./mol  at  64  degrees  F, 
Schuller  r^nd  Wartha: 

Temperature  of  comhustion  32  degrees  F. 

Pressure  of  comhustion  =rl4.7  lb.  per  dq.  in. 

only  as  product  of  combustion 
Saturation  pressure  = 212  degrees  F. 
i/'  ^1151.7  i J = 0 X = 0 

,q/  =18j^1151.7  = 20,731 

,0,^  = 1408 

-Hp  = 20731-1408  - 19,323  B .T.U. 

Hp  (observed)  = 68250 >1.8  = 122852 

Hp  = 122652-19323  =-103529  B.T.U.  per  mol  at  32  degrees  F. 

As  before  (Than  page  97.) 

Hp  (at  62  degrees  F.)  = 103529  ■^74  =103, 603  B .T.U.  per  mol. 

Collecting  results  add  neglecting  the  small  correc- 
tion necessary  to  transfer  the  various  values  to  the  heat  of  com- 
bustion at  62  degrees  F,  we  have  the  following  table  of  values 
for  the  lower  heating  value  of  hydrogen  at  constant  pressurd  and 
at  62  degrees  F. 


102. 


Date 

Investigator 

Value 

1848 

Andrews 

104,597 

1852 

Favre  and  Silhermann 

104,875 

1873 

Thomsen 

103,858 

1877 

3 chillier  and  Wartha 

103,603 

1881 

Than 

103,906 

1883 

Berthelot 

104,962 

1903 

Mixter 

103,034 

1907 

Rumelin 

103,424 

We  shall  omit  the  values  of  Andrev/s,  Favre  and 
Silhermann,  and  Berthelot  as  being  obviously  high.  We  shall 
weigh  the  remaining  five  values  according  to  the  following 
table  f and  take  the  average  as  our  fina,l  result  for  the  lower 
heating  value  of  hydrogen  burned  at  62  degrees  F.  and  at  con- 
stant pressure.  Thomsen  value  is  given  double  weight  be- 

cause of  the  general  accuracy  of  his  experiments  and  because 
his  result  is  the  average  of  seven  experiments  in  which  a total 


of  18  grams  of  water  were  formed,  this  being  a much  larger  amount 
than  that  fomed  in  the  experiments  of  any  of  the  other  investig- 
ators. The  Mixter  and  Ruraelin  values  are  given  the  weight  of 
three  because  of  the  comparatively  recent  dates  at  which  their 
work  was  done.  Also  IIixter*s  value  is  the  average  of  fourteen 
experiments  all  of  which  are  within  1.^  of  the  average  and  Rum- 
elin’s  value  is  the  average  of  7 experiments  all  of  which  are 
within  0.7^  of  the  average. 


Investigator.  Wt.  of  value  in  taking  average.  Value. 

Thomsen  £ 105,858 

Schuller  and  Wartha  1 ;L03  603 

1 103|906 

Mixter  3 103,034 

Rumelin  3 103,424 

Average  l03,459 


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As  our  final  result  we  shall  take  the  rounded  nalue  for  hydrogen 
as  =«  103,450  B .T.U.  per  mol  at  62  degrees  F. 

II.  Carhon  Monoxide. 

7/e  have  the  following  experimental  results  avail- 
able for  the  heating  value  of  carbon  monoxide. 

Date  Investigator.  Value  TemperatU2E 

1848  Andrews  3057  cal/litre  at  cons.vol.  15^0. 

1852  Favre  and  Silbermann  2402.7  cal/gram  ” ” press.  18°C . 

1873  Thomsen  67960  cal/mol  " " ” 18°C. 

1881  Berthelot  68200  " ” " " " 10^0. 

Andrews  and  Berthelot  used  the  bomb  calorimeter. 

Favre  and  Silbermann  and  Thomsen  used  the  same  apparatus  as  they 
did  for  hydrogen  except  that  the  products  of  combustion  were  led 
out  of  the  combustion  chamber  through  a coil  of  small  pipe  of 
considerable  length  which  v;as  immersed  in  the  water  of  the  cal- 
orimeter, thus  insuring  that  the  products  of  combustion  were 
brought  back  to  the  initial  temperature. 

Transferring  these  results  into  meanB.T.U.  per  mol 
at  constant  pressure  using  Callendar’s  specific  heat  ratios  cUid 
neglecting  the  correction  to  62  degrees  F. , we  have  the  followirg 
for  H^at  62  degrees  F. 

Andrews  123,800 

Favre  and  Silbemann 120,900 

Thomsen 122,130 

Berthelot 122,920 

of 

Andrews*  value  cannot  be  considered  because/the 
imperfection  of  his  method.  He  took  only  one  reading  of  the 


• \ 


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104. 

water  temperature  in  his  calorimeter  after  combustion  and  that 
one  just  thirty  seconds  .after  combustion  occurred.  Due  to 
the  design  of  his  apparatus,  the  thermometer  hsd  to  be  removed 
in  order  to  rotate  the  apparatus  which  means  was  used  to  keep 
the  water  at  a uniform  temperature.  To  get  the  temperature 
of  the  water  the  rotation  was  stopped  and  the  thermometer  in- 
serted. It  can  be  seen  that  this  manipulation  is  subject  to 
many  errors. 

Z^^Cathon  monoxide  used  by  Pavre  and  Silbermann  in 
their  experiments  contained  about  zfo  hydrogen  by  weight.  The 
correction  for  the  heat  of  combustion  of  this  hydrogen  content 
amounted  to  about  50%  of  the  heat  resulting  from  the  combustiai 
as  observed  in  any  one  determination.  The  possibility  of 
error  in  this  correction  is  very  great  because  of  the  methods 
of  gas  analysis  in  use  at  that  time  and  also  because  it  automat- 
ically brings  in  all  the  errors  of  their  determination  of  the 
heat  of  combustion  of  hydrogen. 

Of  the  two  remaining  values  we  choose  that  of 
Thomsen^s  in  favor  of  Berthelots  for  the  following  two  reasons. 
First,  for  apy  given  determination  the  volume  of  gas  used  by 
Thomsen  was  about  six  times  that  used  by  Berthelot,  this  tendirjg 
to  reduce  Thomsen's  error.  Second,  Berthelot* s result  is  the 
average  of  five  experiments  where  a total  of  about  1.3  liters 
of  carbon  monoxide  v/ere  burned,  while  Thomsen's  result  is  the 
average  of  10  closely  accordant  experiments  where  in  a total  of 


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alDOut  16  litres  of  carbon  monoxide  were  burned.  The  first  six  oi 
the  experiments  were  performed  in  one  calorimeter  v/hile  the  last 
four  were  performed  in  another  larger  calorimeter.  The  aver- 
age result  of  the  first  group  of  experiments  is  exactly  eq.ua! 
to  the  average  of  the  last  two. 

vYe  have,  therefore,  as  the  heating  value  of  1 mol 
of  carbon  monoxide  at  constant  pressure  at  62  degrees  F.  in 
terras  of  the  mean  B.T.U.  the  following: 

/ 

=122,130 
III.  Methane.  011^ 

Y/e  have  the  following  data  available  on  the  heating 
value  of  methane: 


Date 

Investigator 

Temp 

degrees  G. 

Higher  heating  value 
cal.  per  mol. 

1848 

Andrews 

15 

209,728  constant  vol. 

1852 

Favre  and  Silbermann 

18 

209,000  " press 

1880 

Thomsen 

20 

213,530  ” 

1881 

Berthelot 

18 

212,400  " vol. 

The  methane  used  by  Andrews  was  obtained  from  a 
a stagnant  pool  end  contained  a large  percentage  of  nitrogen 
which  invalidates  his  result.  It  is  also  very  probable  that 
the  Favre  and  Silbermann  value  is  low  because  of  impurities  in 
their  gas. 

Thomsen  generated  his  methane  from  zinc  methyl 

and  hydrochloric  acid,  and  purified  it  by  bubbling  thru  cuprous 

/afcsT 

chloride  solution.  The  above  result  is  from  Thomsen&^work  on 


'i 


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r.'  t 


;.:  ■ 1-:  j ‘is  !!c' 

- .:w-x  ii,.£l  . 

XviXi'  J ..0  0^'  L * j-  ^ . ; , i 10/iilL.^r  ' , . 

*"•  •■  . n*>  '.h.*!  . •• ; r.  ; : 

• ''  iXCi.  .1  • • . 


■-  ' ■ ‘ L .;  -■  X.U6ft*f  0 . • 
' ,C  i}._;:  OJ 

' A *ji(’  - i 

Bi  \Vj(c. 

< or.;  to  ftri.-'v*' 

o?:  ^ ^ f,>  f ^ , H 


ni 


V..  C._,f  .Ill 

f.o  uj' ..iil:.iv..  lii-  otlo  . Li'J  ov* 

: 'L  1?  : A?'  eirXi-v 


■.■  ; L .: 

♦ li' 


■ iPl'T 

:.'0 


- '•rf'  <*  ^ • 'j  o . . G • ^ 

'••'*•.  '•  • 0 , 

T*  tl 


** u - 
_0--.C-. 

ax 

bX 

0: 


i;nj......  .1.1^ 


,.YV3\ 

.r  v-rt'i: 
C'vJl  M.:J  t£-J ' 


fe{-X 

oeox 


e r;fTj-.':  jt.c  ajw  r-YO”  .r^  ? b..,,  c;  j e^lX 


■J-A 


fiv^ -•,.*.  J i . -Ci  L'M.ioHwU'Aw \ , " . ; i.  ^i.;'j:'-.^  :Cr  -r  lui-ti  v i‘. 


• C.‘  \ '.  ; 4 -Jl  . X ' i .i  I 


IX^;:  OY  til.i  i.U’i  -'rili'yr.-il  Y 


I A 


■ ; '1j  Oi  , 00'  ;C  £ oL  c^iTii  v''-XI-  6;r  ■'  tYV.j*';  orf. 


f.  Xr;  c6'S''.i,  cn-.'iU O!  ■ Y'-  '.C'rj  y.^Cv' vii?.  ;. 

CL  -:.*:  ;,):  . -xftX  I i^uX;'.XiiJ.*; , ji'.<  ...  U.yC Ino.:'ii^rf  CO# 

uv  ioiYCV:;^::  _oi.;vXiV  :...  c !:  si  ’-yvofj  y/'t  ..1;,;^  6.YX*io-j^r 


106 


methane  I and  is  the  average  of  nine  experiments  which  show  a 
maximum  variation  of  1.1^.  The  calorimeter  used  was  of  the 
constant  pressure  type,  the  products  of  combustion  being  led 
out  thru  a long  tube  winding  around  the  combustion  ch‘-mber  as 
described  before.  Correction  was  made  for  the  non-condensed 
vapor  in  the  products  of  combustion,  this  correction  being  very- 
small. 

The  Berthelot  result  is  the  average  c£  four  determ- 
inations made  with  his  bomb  calorimeter.  These  four  ex]^ri- 
ments  show  a variation  of  1.6^. 

In  order  to  compare  Thomsen’s  and  Berthelot ’ s values 
we  shall  reduce  them  both  to  the  locker  heating  v^ue  in  metn 
B.T.U.  per  mol  at  62  degrees  P. 

Thomsen:  lie  thane  was  burned  withUie  theoretical 

required  oxygen  at  constant  pressure  of  1 atmosphere  ^nd  temper- 
ature at  62  degrees  P.  Reaction  is  GH^  t 20^ 

Partial  pressure  of  H^jO  vapor  in  products  is 
0 ~ 2/2^14.7  = 9.8  lb.  per  sq.  in. 

Saturation  temperature  at  this  pressure  is  19E  degrees  P. 
i/'  ^1143.9  li  =30.1  X =■  0 

^ 36(1143.90  - 30.1)  = 40,097 
= 2037 

-Hp  = 40,097-2037  » 38,060  B.T.U.  per  mol  CH^ 

Kp  - 213,530  xl. 8 >-0.99842  = 383,7S0  mean  B.T.U.  per  mol. 

Ep  i 383,750-38,060  345 , 690  B .T .U.  per  mol . 


*.  » 


‘I 

I, 

Ij 

I 

i 


f I. » ■ . . i.  ^ ip  A*  V ‘ i V j"  ! . ^ *.i  . T 


t 

I 


I 


I 


I 


•I  <•'  . "1  1. 


V 


I'll. I iK  ‘;,i 


tf 


1 


• J-. 


I 


• i. 


.* 


JU-V< 


J. 


•> 


■* 


f' , 

j , ; 


i 


I 


(■;  O'l 


j!  Lj-  . 


/ 


( 


I 


cf 


.U  A./ 


I 


r V 


/ .1 


i: 


J.  . )U 


a. 


107 


Berthelot:  Methane  was  hurned.  with  excess  oxyg'en 

at  constant  voliome  and.  temperature  of  62  cLegrees  S’. 

Pressure  of  dry  CH^  in  initial  mixture  -1  atm.  -14.7  Ih.  per  sq^. 
in.  Total  initial  pressure = 3.4  atm. (approx) 

Heaction  is 

GH^  i-  2.40^  ->G0^^  2H^0  +0.40^ 

Partial  pressure  of  water  vapor  formed  in  products 
= 2 xi4,7  =.£9.4  lb.  per  sq.  in. 

Partial  pressure  of  water  vapor  required  to  saturate  initial 
gases  at  62  degrees  P.  — 0.276  lb.  per  sq.  in. 

Py^^^  = 29. 4-f  0.275  = 29.675  lb.  per  sq.  in. 

Volume  of  H^O  per  pound  in  products  at  62  degrees  P. 

29. 672  ^ 0^0  —10.475  cu.  ft.  per  lb. 

Saturation  temperature  - 267  degrees  P. 

u/'=  1093.4  uj  -30.1  x-0 

=36(1093.4  - 30.1)  = 38279 
,0,^  =2415 

H'  = 38279  - 2415  =35864 

E’  =212400^1. 8>0. 99842  =381710 

H^^H;,=381710  - 35864  = 345846  B .T .U.  per  mol. 

Thomsen  H 345690 

Berthelot  H 345846 

average  345768 

Lower  heating  value  of  methane  at  62  degrees  P. 

-345770  meanB.T.U.  per  mol. 


I 


I 

• ..  • rU !tfi  •!  t.v  . : . c 'I  . 


•d 


'v 


■It.- 


1*1’  ■ ' Uj  c - ■ .',  lU 


i.,mtTr-r  j-fi- 


fi 

t, 


f- 


. j . r*  -j'  j '»•  ? ; v' ' ■ 


: •'  » . L J.  : 


( C'*:a‘ 


*lu  V - e^Tc  v i i . ; , 


c«  ^ . 


- t- 


V.*  V i)  . '«  , 


•i  J 


J j;n. 
. ' *S' 


',KV  'icJ’.-.r  ^ , 

- ..."  . ■ '••{*£’  r;*  J't;  I Li. 


. *i  *. 


^ . 


~iZ  Ij.  ' 


lA'.'lj  fo  ■ 


r 


IV . Acetylene  ( J 

We  have  the  following  experimental  data  on  the 


108. 


heating  value  of  acetylene: 

Date  Ij^vestigator  Temp.de-  Higher  heating  value 

grees  G.  cal.  per  mol 


1880  Thomsen  19  310050  const,  press. 

1881  Berthelot  18  314900  " vol. 

1906  Mixter  20  311400  " ” 

{ observed  heat 
not  corrected) 

The  methods  used  by  the  above  investigators  are 
the  same  as  described  previously  for  other  gs;ses. 

For  the  lower  heating  values  v/e  have  the  following 

calculations: 

Thomsen:  Acejsylene  burned  with  the  theoreticsl  req.uired 
oxygen  at  constant  pressure  of  one  atmosphere  and  temi:® rature 
of  66  degrees  F.  = 526  degrees  F.  (abs.) 

The  reaction  is  -/-2i-03_ 200^^  +Ha.O 

The  partial  pressure  of  H^O  vapor  in  products  is 
= 1/3 ->^14. 7 =4.9  lb.  per  sq..  in. 

The  saturation  temperature  at  this  pressure  is  161  degrees  F. 
i/'  =1131.2  i/  =34.08  x = 0 

,0.1  = 18(1131.2  - 34.1)  = 19748  B.T.U.  per  mol  of  C^H  ^ 
,0^  = 742 

= 19748  - 742  =19006  B.T.U.  per  mol. 

= 310050^1.8^0.99842  =557200  B.T.U.  per  mol 

= 557200  - 19000  = 538200  B.T.U.  per  mol. 


I 


r«. 


0^ 


■ ^ /*w^  .-*»  *.  ♦»-  f * t *\  .-*^ 


♦ L 


L 


*u«:. 


U.  ..  >>.  ■.  :'■  V • i.  ipto 


-t.  . ' 


I . 


.t 


f\ 


f I- 


L'C  :/ 


’.If,) 

I 


H' 


' ■ . "i 


^.v;  r; 


<yx\.x  - 


\ 


jT 


109, 


Berthelot:  Acetylene  burned  at  constant  volume  with  excess 

oxygen  at  temperature  of  62  degrees  F, 

Pressure  of  dryC^H^in  initial  mixture  —one  atmosphere 
*14*7  Ih . per  sq.  in. 

Reaction  is  ■*”  2G0j^  -f-  + 

Partial  pressure  of  water  vax:or  required  to  saturate  initi?!. 
gases  at  62  degrees  P.  =■  0.275  Ih.  per  sq.  in. 

P^^^  = 14.740.275  ==14.975 

Volume  of  HJ)  per  pound  in  products  at  62  degrees  F. 

_ 14.7  ^ 580.6 

“ 14.975  18  “ 2a756 

Saturation  temperature  for  this  volume  is  194  degrees  F.  = 654:FaJ>j, 
u/'  - 1075.5  u^'  = 50.1  x:^  e 

= 18(1075.5-50.1)  = 18781 

- V71 

=18781  - 771  = 18010 

514900> 0.99842^1.8  = 565920  B.T.U.  per  mol. 

- 565920-18010  = 547910 

Hp  ~ 547910  -*-522  =548452  B.T.U.  per  inol. 

Mixter:  Initial  pressure  of  acetylene  =14.65  lb.  per  sq.  in. 
Initial  gases  dry  at  20degrees  C . = 68  degrees  F. 

Therefore,  final  pressure  of  wtter  vapor  considered  non-condensed 
= 14.65  lb.  per  sq.  in. 

Volume  of  1 lb.  of  water  vapor  in  products  of  combustion 


. 580.6  .528  ..14.70 

IS" 


21.46 


no. 

Saturation  temperature  = 224  degrees  F (from  steam  tables) 
u/'  a 1082.2  u'  =56.1  ^=  890.2 

Saturation  pressure  at  T =68  degrees  P.  “ 0.539  lb.  per  sq.iru 
X ^ 0.339/14.65  = 0.0231 

-f-  = 18459 

,Q;.=915 

hJ  18459-915  = 17544 

(observed)  = 311400 XI. 8X). 9 9842  - 559630 
- 559630-17544  - 542086 
Hp  = 542086  +528  = 542614  B.T.  U /mof 

Collecting  results  we  have  for  the  lower  heating  value  of 


acetylene  at  constant  pressure  and  62  degrees  P.: 


Thomsen  538,200 

Berthelot  548,432 

Mixter  542 , 614 

Weighing  Thomsen's  value  2,  Berthelot' s 1,  and  Mixter' s 3,  we 
have  as  the  average 

= 542,110  B.T.U.  per  mol.  at  constant  pressure  and  62°  P. 


Y.  Pthylene  (C,Hy) 

Experimental  data  on  the  heating  value  of  ethylene: 


Date  Investigator 


Temp,  de-  Higher  heating 

gfees  C.  value  Cal. per  mol. 


1880 

1881 

1901 


Thomsen  17.9 

Berthelot-llatignon  16.8 
Mixter  18.8  (Obs.) 


333350  const,  press. 
340000  " vol. 

345080 


Por  the  lower  heating  values  we  have  the  folloMng 

calculations : 


-It 


f 

:i 


i; 


Hn$% 


-O 


. -.Cl  .- 

ncijo-ri/viie 


» . • r*  • ^ 


iF.VI  .-  ^ -.:  ;v'.»;  . . - 

.-M.  , .'  n x: :;;  (t.v..cUoc> ) 

aeoHKif’^-  k 


’IdV  ■ \ .‘Cl  J‘i:  ,,.-  .(  ^v/ 


t -.,  t > ' \ 

■-  . i .-i  j 


■>-.;'.  V->  • 

•i‘J-.ct£  ©O  -J  i’fU’S^i  SlXO 

j v' 

.V  ■-•^  /.if- 

'I  *' 

f ■ •* • ^,**rwix  -4’ 

•*  '-J 

. - 0 

■'  ■ 
-■yn 

r - ' * 


TfriJ 


!:  'nee:  c,‘-  J.ilr4 Xl. 
c-’^>,.;  'pv  . c ' ::  evn'i 


fii!  ?•*  i;BtiC't'  ‘ ('  Cc  0 0 


101  • ■ * ^ r,  5^  i * 

, ' ■'■-•?' Sffi 


;w* ' 'T 

• • 

:oro 

'■  ../tV 

I-ja  0/; 

i \i  s .'v  X J.  »#  - u 

..  ^ ■ ■'i*" 

^ • 4.  A ^ 

i.-i  ^--ii 

l.;/r''':iH 

— C 

. rUe  '. 

',0'.  .;T  • 

OJu  . 

. : . A) 

r -r  vAv  v 

r 

• • 

*'  . . * ^.fT  C 

■*  ‘ ■ *>  -/-A*  ,v.' 

C80X 

-• 

f .-j^y 

;.^X  r-or:.'.^’  :.' 

Ibex 

. ':r,ix: 

. C 

lis'iSfxr 

xoex 

• . >'  ‘ 

'■'.A' 

f*. '-X  **  r i t ' 

9r  :fv  'j 

m^ujLMy 

■Afll.OSOi!  1€S»C'i7 

: inX..  .''  riitc, 


Ill 


Thomsen:  Ethylene  burned  vath  the  theoretical  amountof 
oxygen  at  a pressure  of  one  atmosphere  =14.7  lb.  per  sq..  in^  and 
temperature  of  64  degreeB  E.  Reaction  is  1 30^ 2C 0 ^ 2H ^0 

Partial  pressure  of  H»i)  vapor  in  products  considered  j^on- 
condensed  is  ‘i'-^l^.V  =7.35  lb.  per  sq..  in. 

Saturation  temp»  at  7.35  lb.  per  sc[.  in.  = 179  degrees  E. 
i,"  = 1138.7  =32.1  X - 0 

-36(1138.7-32.1)  = 39838 

= 1800 

= 39838  - 1800  = 38038  B.T.U.  per  mol  C^H  ^ 

=333350^1.8x0.99842  =599070 

Ep  = 599070  - 38038  = 561030  B.T.U.  per  mol 

Berthelot:  Ethylene  burned  at  constant  volume  with  excess 

of  oxygen  T = 62  degrees  E. 

Pressure  of  dry  C^Hy.in  initial  mixture  -1  atm.  -^14.7  lb  .per 

sq.  in. 

3^^  200^-f-  2Ej:>-hiO^ 

Partial  pressure  of  vapor  in  products  of  combustion 

considered  non-condensed  = 2 >14.7  =29.4  lb.  per  sq.  in. 

P:rtial  pressure  of  water  vapor  required  to  saturate  initial 

gases  at  62  degrees  E.  = 0.275  lb.  per  sq.  in. 

P^  ^ = 29.4  +0.275  = 29.675  lb.  per  sq.  in. 

Volume  of  H^O  per  pound  in  products  at  62  degrees  E. 

14 .7  ^ 380 .6  A tn  r-  O j_  -1 

29  672  q0  ~ 10.475  cu.  ft.  per  lb. 

Saturation  temperature  »•  267  degrees  E. 
u;  - 1093.4  u»  =30.1  x=0 

iQa^  = 36(1093.4-30.1)  = 38279  B.T.U.  per  mol 


T 


ce’-(- 


J-  • 

*.  j ■ ir.  i[4io  J 


!.i  mi  ^ 


» . •/' 

0 -s  X 


uiir-  o • 


t \ 9 ^ 


,£i  .1 

CK'OX  - 


ir%  r 

^ ••  ■ 


CO 


■~ 


u. 


, .1 


: A 


lL  'J.r  fTi..:, 


j; 


..V  * . 


• '>018*'  '■'  '*  - 

■ > .-.  / ' i ' j .,j  ^ . 

, T I,  . .;..  , * C 
‘ -r.  £ ]:•:..<! 

• • . ..  J,  . c 5: 'iJj;'.  (,  - ■ .u'l  ij .i'r  tiii't'. 

■.c)<;;iV  ioi,'i  '.  : •:  I £•  ‘->'7 


,.»  * , » • »n 

VI  ^ . V w*  . « • 


0> 


.'C  l 


^ ' ' L> ; 


• * r*  ^ 

, 4.  . 


V 


^ u • 


X 


n *^^ 


ToJ".  tiolvi  *11/448  /.J 


. /;. 


X **  ;xy 


i:-f''  *;  '.J 


. < *’  C7  ■ * ' 


112. 

= 2416 

hJ  ^38279  - 2415  « 35864 

Ey  ^ 340000’^!. 8*0. 99962  =-  611760 

Ey  ^ 611760  - 35864  - 576896 

= H ^ = 575896 

Mixter;  Initial  pressure  of  ethylene  *14.7  Ih.  per  scl-  in. 
at  18.8  degrees  C.  - 66  degrees  I’.  Initial  gases  dry. 

Fin>.l  pressure  of  water  vapor  considered  non-condensed 
» 2*14.7  =•  29.4  Ih  . per  sq..  in. 

Volume  of  1 Ih.  of  v/ater  vapor  in  rroduots  of  coiibustion 
oonsi  'dBred  non-condensed  III  ^ -10.653  cu.  ft. 

Saturation  temperature » 266  degrees  P.  (from  steam  t ahles) 
u/  =1093.1  u/  ^34.1  858.8 

Saturation  pressure  at  T - 66  degrees  P =0.316  To.  per  sq.in. 
X = 0.316/29.4  = 0.0011 

,Q'=36  0.093.1  - ( 34.1  to. 0011*858. 8)J  - 38092S.T.U.per  mol 
,Q  ^ = 2360 

h'  -Ey  c 28092  - 2360  =35732 
hJ  = 345080*1.8*0.99842  = 620160 
Hv  =Hp  =620160  - 35732  =584430 

Collecting  results  we  have  for  the  lower  heating  vd.ue  of  ethylene 
(GjH^)  per  mol: 

Thomsen  561,030  B.T.U.  per  mol 

Berthelot  575,900 
Mixter  584,430 

.Veighing  Thomsen’s  value  2,  Berthelot’s  1,  and  Mixter’ s 3,  we 
h<-ve  the  following  value  for  the  lov/er  heating  value  of 


113 


ethylene  at  6E  degrees  F.at.  constant  pressure: 
H^z:.575,210  B.T.U.  per  mol. 


VI.  Ethane 

Esp  er imen tal  value s : 

Date  Investigator 


1893 

1905 


Berthelot 

Thomsen 


Temp. de- 
grees C . 

IS 

18 


Heating  value (higher) 
Gal.  per  mol. 

370,900  const,  volume 


370.440 


pressure 


Heduction  to  lower  heating  value: 

Berthelot:  Ethane  Burned  at  constant  volume  with 

excess  oxygen  at  temperature  of  55  degrees  E. 

Pressure  of  dry  in  original  mixture  -s-l  atm.  -14.7  lb.  per 

sc[.  inch. 

Reaction  is  ECO^  3H^0  y- -g-O^, 

Partial  pressure  of  water  vapor  formed  in  products  considered 
non-condansed  is  3 a14.7  =44.1  lb.  per  sq.  in. 

Partial  pressure  of  water  vspor  required  to  saturate  initial 
gases  at  55  degrees  P.  = 0.E14  lb.  per  sq.  in. 

P^^  = 44.1 +0.E14  =44.314  lb.  per  sq.  in. 

Volume  of  H^O  per  pound  in  products  at  55  degrees  F. 


380.6.515  ,14.7 

X." 


= 6.9S0  cu.  ft. 


18  5EE  ^44.314 
Saturation  temperature  = E95  degrees  F.  (steam  tables) 

u,"'  =1099.8  u^'  = E3.1  X = 0 

= 54(1099.8  - E3.1)  = 58.14E 

,0,^  = 4E60 


' I 


I . 

I' 


'j 

/ 


t 


•r 


' V 


r 


r 

w • 


r ■ t 
»v#r»rr  f • : J. 


I 

i 

^ ' I 


J 


.i 


r 

-I.  / cj 


r 


114 


hJ  - 58142  - 4260  - 53,882 
H;  =370,900^1.8^0.99962  = 667,360 

hJ  -667,360  - 53,882  = 613,478 

1-T  = 613,478  1 515  = 613,993 

Thomsen:  3thane  hurned  at  constant  pressure  of  1 atm. -14. 7 
Ih.  per  sq.  in,  with  the  theoretical  amount  of  oxygen  at  18  de- 
grees G . = 64  degrees  P.  Reaction  is 

f 3-|0^  2C0,.'H  2H^0 

Partial  pressure  of  water  vapor  in  products  considered  non-con- 
densed  is 

= 3/5-'14.7  - 8.81  Ih.  per  sq.  in. 

Saturation  temperature  at  8.81  Ih.  per  sq.  in.  ' 187  degrees  F 
i/'  = 1141.9  = 32.1  X - 0 

-54(1141.9-32.1)  = 59,929  B.T.U.  per  mol 
= 3 [7.03(647-524)  i-k  1.25 -lO'^  ( 647*’  -524^)  l/S- 0.2(  647^ 


= 2892 

-Hp  = 59,929  - 2892  - 57,037 
Hp  370,440^1.8x0.99842  - 665,730 
^665,730  - 57,037  - 608,693 

Results  for  the  lov/er  heating  value  of  ethane  (C^H^)  at  constant 
pressure  and  62  degrees  F 


Weighing  Thomsen’s  value  2,  and  Berthalot’ s 1,  we  get  for  the 
average  lower  heating  value 

= 610,460  B.T.U.  per  molat  62  degrees  P.  and  const. press. 


Thomsen 


608,693 


Berthelot  613,993 


N, 


•♦t 


V 


ii, 


>rU 


■ 1 ’ -‘■ 


a j 


I , 


w • 


. I 


.f  -'*>i 


-J'  : : 


J ■/ 


u , 1 CJ; 


C‘:  '‘.i  j 


I J 


115* 


VII.  Benzene  vapor:  (O^H^) 

For  the  heat  of  oomhustion  of  Benzene  vapor  the  ex- 
perimental value  of  Stohman,  Rodatz  and  Herzherger  v/hioh  is 
10,096  cal.  per  gram  at  17  degrees  G,  is  chosen  as  the  Best 
available.  It  is  the  average  of  12  experiments.  The  method 
used  was  to  pass  a current  of  air  over  a wad  of  cotten  saturated 
with  Benzene  liquid  and  Burning  the  resulting  mixture  of  v^or 
and  air  in  a constant  pressure  calorimeter.  The  products  of 
combustion  v/ere  ledd  through  a long  spiral  tube  and  then  thru 
absorbers  to  remove  the  moisture  in  the  usual  way. 

Calculation  of  the  lower  heating  value:  We  shall 

assume  that  the  oxygen  and  benzene  in  the  benzene -air  mixture 
are  present  in  the  theoretical  proportions  for  combustion.  The 
reaction  equation  is  C^  -/-7.5^^  -h  S8.6H^  600^  -t  3EJ)^80  5M^ 

The  partial  pressure  of  the  benzene  vapor  in  the  original  mixture 
is  1/37  X14.7  0.40  lb.  per  sq.  in.  According  to  Young* 

the  saturation  pressure  of  benzene  vapor  at  17  degrees  C.  is 
65  mm.  Hg,  or  1.25  lb.  per  sq.  in.  Since  the  assumed  partial 
pressure  of  the  benzene  vapor  in  the  initial  mixture  is  only 
about  one  third  of  the  saturation  pressure  of  benzene, at  17  de- 
grees C.,  the  assumed  partial  pressure  can  be  easily  attained 
and  is  reasonable* 

The  partial  pressure  of  the  water  vapor  in  the 
products  of  combustion  considered  non-condensed  is 
= 3/37.5^14.7  = 1.176  lb.  per  sq.  in. 

*Young.  Scientific  Prodeedings  Royal  Dublin  Society. 

Series  2.  v.l2.  p.422.  (1910) 


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'• 


116 


Saturation  temperature  at  this  pressure  is  107  degrees  P. 
i/' *1107.7  =-30.1  (at  17  degrees  C.  62  degrees 

,Q,  ? 54(1107.7  - 30.1)  ® 58190  B.T.U.  per  mol  benzene. 

*1,050 

-Hp  = 58,190  - 1050  = 57140 
Ep  = 78.06*10,096*1.8^0.99842  = 1,416,460 

Ep  « 1,416,460  - 57,140  1,359,320  B.T.U. per  mol  at  62^  P. 

VIII.  Amorphous  Carbon 

vVe  have  the  following  experimental  datasvailable  for 

the  heating  value  of  amorphous  carbon: 

Date  Investigator.  Value,  cal.  per  gram. 

1848  Andrews  7,678 

1852  Pavre  and  Silbermann  8,080 

1883  Gottlieb  8,033 

1889  Berthelot  8,137 

Andrews  states  in  his  discussion  of  the  value  given 
above,  which  is  the  average  of  eight  determinations  with  highly 
pureified  wood  charcoal  in  the  bomb  calorimeter,  that  in  spite 
of  the  presence  of  excess  oisygen,  carbon  monoxide  was  found  in 
the  products  of  combustion.  This  fact,  of  course,  renders  his 
value  too  low. 

Pavre  and  Silbermann  also  found  carbon  monoxide  in 
the  products  of  combustion  from  their  experiments.  After  de- 
termining the  amount  of  carbon  monoxide  present  in  any  one  nase 
they  added  to  the  observed  result  the  heat  of  combustion  oJl  this 
given  amoujit  of  carbon  monoxide  so  that  their  final  results 
give  the  heat  of  combustion  of  carbon  to  carbon  dioxide.  This 


I, 


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117 


correction  in  their  case  amoimts  to  only  about  three  percent,  so 
that  errors  introduced  by  using  an  incorrect  value  for  the  heat 
of  combustion  of  carbon  monoxide  are  insignif icant.  Eighteen 
experiments  in  three  series  were  run  using  highly  purified  vnod 
charcoal.  The  values  of  the  first  series  consisting  of  five  ex- 
periments showed  a maximum  variation  of  89  calories  in  the  values 
given  for  the  heat  of  combustion  per  gram.  The  average  value 
from  the  first  series  is  8,086  calories  per  gram.  The  next 
seven  experiments  constituting  the  second  series  showed  a maximum 
variation  of  31  calories.  The  average  of  the  second  series  is 
8,081  calories  per  grrrnn.  In  the  last  six  experiments  wood  char- 
coal purified  in  different  ways  was  used  in  different  determina- 
tions to  note  if  the  method  of  purification  had  any  effect  on 
the  results.  In  this  last  series  the  maximum  variation  between 

any  tv/o  results  v/as  19  calories  and  the  average  of  these  six  was 
8,080  calories  per  gram.  This  is  the  result  q^uoted  above. 

Gottlieb  used  a calorimeter  very  similar  to  tin  one 
used  by  Eavre  and  Silbermann.  The  carbon  used  b^  Gottlieb  was 
prepared  by  heating  a five -gram  ball  of  cotton  in  a loosely  cov- 
ered dish  alov/ly  at  first  and  then  more  intensely  after  all  the 
combustible  gases  had  been  driven  off.  Later  the  cotton  char- 
coal was  transferred  to  a tightly  covered  platinum  dish  and  neated 
to  about  950  degrees  C.  for  some  hours  and  then  cooled  in  a des- 
sicator.  This  carbon  absorbed  moisture  freely.  Upon  analysis 
the  sample  showed  1.5^  moisture.  It  is  safe  to  assume  that  only 
a portion  of  this  moisture  was  absorbed  Iby  the  time  the  sample 
for  combustion  was  weighed  end  the  rest  was  absorbed  during  the 


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118 


timethe  calorimeter  was  being  prepared  for  operation*  With  a 

slight  amount  of  moisture  present  in  the  sample  as  weighed,  of 

course,  the  final  result  oalcu-lated  on  the  basis  of  this  weight 
% 

will  be  slightly  low.  Therefore,  Gottlieb’s  result  ^ich  is  the 
average  of  six  experiments  which  have  a maximum  variation  of  7 
calories  per  gram  points  to  the  accuracy  of  the  Pavre  and  Silber- 
mann  value* 

Berthelot’s  value  was  obtained  by  burning  wood  char- 
coal, very  carefully  purified  and  dried.  Oxygen  under  25  atmos- 
pheres pressure  was  used  in1he  bomb  calorimeter  to  insure  com- 
plete combustion.  The  value  given  above  is  the  average  of  six 
experiments  which  show  a maximum  variation  of  10  calories  per  gram- 

Of  the  above  values  quoted  we  shall  choose  the  Pavre 
and  Silbemann  value  which  is  substantiated  by  a total  of  E4  ex- 
periments in  preference  to  that  of  Berthelot  vii  ich  is  the  result 
of  only  six  experiments.  Also,  it  is  very  possible  that  Berthelot’ £ 
method  of  purification  was  at  fault. 

Seducing  the  Pavre  and  Silbermann  value,  which  is  in 
terms  of  the  20  degree  calorie,  to  meanB.T.Ui  per  mol, we  have 
for  the  heating  value  of  amorphous  carbon  at  62  degrees  P: 

H - 174,250  B.T.  U*  per  mol  at  62  degrees  Pahrenheit* 


119* 


Talile  of  Heats  of  OomlDUStion 


No*  Reaction  Lower  heat  of  comhustion  at 

constant  pressure  and  62  de- 
terees  in  mean  B«T.U. 


per  mol. 

per  Ih. 

per  cu.ft. 

1. 

103,450 

51,725 

271.8 

2. 

G0-+i0,.-f  co^ 

122,130 

4,361.8 

320.9 

3. 

H^O+GO 

-18,680 

— 

4. 

G ( arnorpjf  GQ2_  -f  2G0 

-70,010 

— 

5* 

C " +0^->CQ^ 

174,250 

14,521 

— 

6* 

C " +2H^-^GHy 

35,380 

— 

7. 

GH^  -t  20,,-^G0^i-2H^0 

345,770 

21,568 

908.5 

8. 

Gj,H^  -t  2-|0  200^  + E^O 

542,110 

20,838 

1,424.4 

9. 

t 30,,  ~^2G0^  1-  EH^O 

575,210 

20,520 

1,511.4 

10. 

G^Ht  -f3iO^-f  2G0^t3H^0 

610,460 

20,316 

1,604.0 

11. 

{ vapor ) +7-JO^->6GO^ -fSE^O 

1,359,320 

17,415 

3;571.5 

' ■ vTri 

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Table  of  Heats  of  Combustion  - continued 


120* 


Ho.*  Lower  heat  of  combustion  at  Heat  of  combustion 

constant  volume  and  62  de-  at  absolute  zero  in 

grees  F.  in  mean  B.T.U.  mean  B.T.U.  per  mol. 


Per  mol 

Per  lb. 

Per  cu.ft. 

1. 

102,930 

51,564 

270.4 

102,100 

2. 

121,610 

4’343.2 

319.5 

121,250 

3. 

-18,680 

-19,150 

4. 

-68,970 

-68,430 

5. 

174,250 

14,521 

174,060 

6. 

34,340 

31,250 

7. 

345,770 

21,568 

908.5 

347,020 

8. 

541,590 

20,838 

1,423.0 

541,600 

9. 

575,210 

20,520 

1,511.4 

576,730 

10. 

610,980 

20,333 

1,605.3 

613,520 

11. 

1,359,840 

17,423 

3,572.9 

1,364,370 

* For 

reactions  corresponding  to  these 

numbers  see  table  on  p.ll9 

i 


1 1 


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L. 


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hi 


fiCHrt*  .»«*«  * 4, 


121 


6.  Berthelot. 


References  on  Heats  of  Comtustion. 

1*  Andrews.  Phil  Mag  (S)  32,  321  (1848) 

2.  Pavre  & Silhermann.Ann  de  Chem  et  de  Phys  (3)  34,  349  (1852) 

3.  Thomsen.  (H^)  Pogg  Ann  148,  368  (1873) 

(CO)  Thermochem  Unters,Vol  II,  p284 
(CHyi )Berichte  d.d.  Chem  Gesell  13,  1523(1880 
(C,_H^)  and  (C^H^)  Thaermo  Unters.vIV,p65. 
(Cj_H^)  Zeit  Physical  Chem.  v.51,  p657  (1905) 
4*  Schuller  & Wartha.  Wied  Ann  2,  359  (1877) 

5.  Than.  Wied  Ann  13,  84  (1861) 

" " 14,  422  (1881) 

(Hj  Compt  Rendus  116,  1333  (1893) 

(C)  Ann  de  Chem  et  de  Phys  (6)18,  89  (1889) 
(CHy)  " " " ” " " (5)  23,  176  (1881 

(C^H^)and  (C^H^)  ” " ” (6)  V.30,p556(l893 

Ann  de  Chem  et  Phys  (6)  v30,p547(l893 
(HiP)  Am  Jour  Sci  (4)  16,  214  (190») 

(C^H,.)  " " " (4)  22,  pl7  (1906) 

(C,_Hy)  " " " (4)  12,  347  (1901) 

Zeit  Phys  Chem  58,  456  (1907) 

Jour  Praht  Chem  28,  420  (1883) 

10.  Stohman  Rodatz  Herzherger.  Jour  Praht  Chem  (2)  33,  257  (1886) 

11.  Callendar  (Specific  Heat  of  Water)  Phil  Trans,  V.212A,  ppl-3^ 

(1913) 


7.  Mixter. 


8.  Rumelin. 

9.  Gottlieb. 


Appendix  C 
Chemical  Eq^uilihriiim 


122 


A hrief  outline  of  the  methods  used  in  determining 
the  eq.uilihrium  composition  resulting  from  gaseous  reactions  at 
high  temperature  is  given  in  the  follov/ing.  The  experimental 
data  used  in  determining  the  thermodynamically  indeterminate  con- 
stant of  integration  in  the  equilibrium  equation  is  given  for 
each  of  the  reactions  studied.  Eor  detailed  descriptions  and 
discussions  of  the  e^q^erimental  methods  see  Haber’s  "Thermody- 
namics of  Teahnical  Gas  Reactions"  and  Hernst’s  "Theoretical 
ChemistryV 

I.  Streaming  Method.  The  gases  involved  in  the  re- 
action are  passed  through  a tube  a section  of  which  is  heated  to 
the  desired  temperature  and  the  following  section  kept  at  a low 
temperature.  The  gases  are  assumed  to  attain  equilibrium  in  the 
hot  portion  of  the  tube  and  to  be  cooled  so  rapidly  in  the  cold 
portion  of  the  tube  that  the  reaction  immediately  stops.  The 
equilibrium  composition  of  the  gases  at  the  high  tenrperature 
thus  exists  in  the  cooled  gases  which  can  then  be  easily 
analyzed. 

II.  Semipermeable  Membrane  Method.  A vessel  which 
is  pemeable  to  one  constituent  only  of  the  reaction  is  evacu- 
ated and  the  reacting  mixture  of  gases  at  the  desired  temperature 
caused  to  circulate  around  the  outside  of  the  vessel.  The  par- 
tial pressure  of  the  one  constituent  to  which  the  vessel  is 
permeable  will  soon  exist  inside  the  vessel  and  can  be  measured 
with  a manometer.  This  method  is  applicable  only  to  a study  of 


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123 


the  dissociation  of  H^O  since  semipermeahle  substances  are  known 
for  hydrogen  only  and  are  palladium,  platinum,  and  iridiiira. 

III.  Maximum  Explosion  Pressure  Method.  The  origi- 
nal mixture  of  gases  is  exploded  in  a closed  vessel  and  the  maxi- 
mum pressure  of  e2qplosion  measured.  Since  the  maximum  pressure 
is  dependant  on  the  heat  of  reaction  and  the  heat  of  reaction  on 
the  ecjuilibrium  composition  the  equilibrium  composition  lends  it- 
self to  calculation  from  the  maximum  pressure.  Also  in  those 
reactions  which  incur  a change  in  the  number  of  mols  the  maxi- 
mum pressure  is  directly  influenced  by  the  equilibrium  composition. 

IV.  Method  of  the  Heated  Catalyst,  (a)  If  a catalyst 
such  as  a platinum  wire  is  heated  electrically  in  an  atmosphere 
of  gas  the  equilibrium  composition  of  the  gas,  at  the  temperature 
of  the  wire,  will  exist  in  the  gas  immediately  surrounding  the 
wire.  Due  to  the  circulation  of  the  gas  set  up  by  the  heated  wire 
the  gas  in  contact  with  the  wire  will  be  swept  iiito  the  cooler  re- 
'gions  and  the  reaction  thereby  "frozen”.  This  process  is  allowed 
to  continue  until  the  composition  of  the  whole  gas  volume  becomes 
constant.  This  condition  is  determined  by  analyzing  samples  of 
gas  from  time  to  time.  Temperatures  are  determined  by  the  change 
in  the  electrical  resistance  of  the  wire. 

(b)  A variation  of  this  method  is  to  heat  a vessel 
containing  the  catalyst  to  the  desired  temperature  and  then  to 
lead  the  gases  through  the  vessel  or  enclose  them  in  the  vessel 
until  equilibrium  is  established.  Samples  are  drawn  out  from 
time  to  time  and  analyzed. 

V.  Iridium  Dust  Method.  Iridium  dust  is  produced 
by  heating  strips  of  iridium  electrically  in  various  gases.  The 


3 


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quantity  of  dust  produced  is  dependant  on  the  nature  of  the  gas 
and  the  temperature  to  which  the  iridium  is  heated.  This  method 
is  especially  applicable  to  the  measurement  of  the  dissociation 
of  carbon  dioxide.  Nitrogen  and  pure  carbon  monoxide  produce  no 
appreciable  amount  of  dust  v/hile  oxygen  produces  large  quantities 
of  dust.  The  amount  of  dust  produced  by  carbon  dioxide  at  a given 
temperature  of  the  iridium  and  at  atmospheric  pressure  is  assumed 
to  be  due  to  the  oxygen  liberated  by  the  dissociation  of  the  car- 
bon dioxide.  A mixture  of  nitrogen  and  oxygen  is  found  v/hich  mil 
produce  the  same  amount  of  dust  as  the  carbon  dioxide  xtnder  the 
same  conditions.  Assuming  that  the  oxygen  content  of  the  tvjo 
gases  is  the  same  the  equilibrium  composition  of  the  carbon  di- 
oxide reaction  is  then  known. 

VI.  Measurement  of  Equilibrium  in  the  Bunsen  Flame. 
The  inner  and  outer  cones  of  the  Bunsen  flame  are  separated  by 
fitting  a glass  tube  as  an  extension  on  the  end  of  the  burner, 
the  glass  tube  being  of  some  what  larger  diameter  than  that  of 
the  burner.  A stopper  is  made  to  fit  tightly  in  the  annular  space 
betv/een  the  burner  and  the  glass  tube.  The  inner  cone  then  burns 
on  the  end  of  the  regular  Bunsen  burner  tube  which  is  now  inside 
the  glass  tube  while  the  outer  cone  burns  on  the  end  of  the  glass 
tube.  Samples  of  gas  are  withdrawn  from  the  space  between  1he 
two  cones  and  are  assumed  to  be  in  equilibrium.  Temperatures  are 
measured  with  thermo-couples. 

VII.  Direct  Detenaination  of  Equilibrium,  (a)  Yi/hen 
one  of  the  constituents  in  the  reaction  is  a solid  such  as  carbon 
this  can  be  placed  in  a porcelain  tube  and  the  whole  heated  in  an 


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electric  furnace.  The  gas  constituents  are  then  passed  thru  the 
porcelain  tube  at  a rate  sufficiently  slov/  to  give  time  for  equi- 
librium to  be  established.  The  products  are  then  analyzed. 

(b)  Carbon  in  the  form  of  rods  is  heated  by  an  elec- 
tric current  in  an  atmosphere  of  the  gas.  Reaction  occurs  at  the 
surface  of  the  rod  and  is  "frozen”  when  the  reacting  gases  diffuse 
into  the  colder  regions. 

In  either  method  YII  (a)  or  (b)  a catalytic  agent 
such  as  platinum,  nickle  or  cobalt  may  or  may  not  be  used. 

(c)  A study  of  the  water  gas  equilibrium  is  made  by 
passing  water  vapor  over  glov/ing  coals  and  analyzing  the  result- 
ing gases. 

VIII.  Equilibrium  from  Density  Measurement.  The  dis- 
sociation of  carbon  dioxide  can  be  measured  by  electrically  heat- 
ing a platinum  bulb  filled  with  carbon  dioxide  to  a desired  tem- 
perature and  then  dropping  into  the  bulb  a small  piece  of  alumi- 
num. Changes  of  volume  are  measured  by  the  movement  of  a short 
thread  of  mercury  in  a horizontal  capillary  outlet  tube  'fitted  to 
the  platinum  bulb.  The  following  reaction  occurs: 

2A1  + AljOy  t SCO 

According  to  this  reaction  no  change  in  volume  will  occur  if  the 

carbon  dioxide  is  undissociated  and  if  the  assumption  is  made  t^at 

equal  v/eights  of  carbon  monoxide  and  carbon  dioxide  have  the  same 

volumes  under  similar  conditions  of  pressure  and  temperature.  If 

the  carbon  dioxide  is  dissociated,  a change  in  the  number  of  mols 

v/ill  occur  with  the  above  reaction  which  is  proportional  to  the 

amount  of  dissociation  and  which  can  be  measured  by  the  change 
of  volume. 


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The  general  expression  representing  the  eq.uilihri'um 
state  in  any  reaction  is  as  follows: 

4.571  log,^Kp=*^  ~£.3026q^lOg^^  T-i  ^T-I/S^'t'^C 

i 

V«here  T"is  equal  to  the  sum  of  the  constant  terms  in  the  instan- 
taneous specific  heat  equations  for  constant  pressure  of  the  con- 
stituents of  the  original  mixture  minus  the  sum  of  the  constant 
terms  of  the  instantaneous  specific  heat  equations  for  constant 
pressure  of  the  constituents  of  the  products  of  the  reaction. 

_ II 

Likewise  ^ is  the  summation  of  the  coefficients  of  T in  the  in- 
stantaneous  specific  heat  equations  while  ^is  the  summation  of 
the  coefficients  of  T’"  in  the  instantaneous  specific  heat  equa- 
tions. 

The  general  expression  for  the  heat  of  reaction  at 
absolute  zero  is 

= Hp-  Tj^'T  + i-r  T / J 

Vi/here  T is  the  absolute  temperature  at  which  is  determined. 

Reaction 
CO  CO^ 

Inserting  the  proper  constants  in  the  above  general  equilibrium 
equation  we  have 

4.571  Iog,/^:=:  - 5.3881  log,^Ti-0.001335T-0.08'10'^  T+C 

Z - (_x)  (3-x)'^.il)^ 

100  ( 1-x)  - percent  dissociation  of  CO^ 

Papressure  in  lbs.  per  sq.  ft.  abs. 

T = temperature  Pahr  deg^’  abs. 


A 


1 27 


Experimental  Data 

Pressure  atmosphere  -2116.7  lb.  per  sq..  ft. 
Temperature 


Do. 

C atbs 

E abs 

100  (1-x) 

4.571  log,^^ 

Method 

Investigator 

1 

1300 

2340 

0.00414 

23.1374 

I 

Dernst  and 

2 

1395 

2511 

0.0142 

19.4673 

IV 

von  Wart enb erg 
Langmuir 

3 

1400 

2520 

0.01-0.02 

19.3040 

I 

llernst  and 

4 

1443 

2597 

0.025 

17.7830 

IV 

von  Yv  art  enb  erg 
Langmuir 

5 

1478 

2660 

0.029-0.035  17.0478 

I 

Dernst  and 

6 

1481 

2666 

0.0281 

17.4361 

IV 

von  Wartenberg 
Langmuir 

7 

1498 

2696 

0.0471 

15.9030 

IV 

Langmuir 

8 

1565 

2817 

0.064 

14.9835 

IV 

Langmuir 

9 

1818 

3272 

0.45 

9.1254 

VIII 

Lowenstein 

10 

2243 

4037 

4.50 

2.2520 

V 

Bmich 

11 

2423 

4361 

10.50 

-0.3710 

V 

Emich 

12 

2640 

4752 

21.00 

-2.6343 

III 

B jerrum 

13 

2879 

5182 

51.70 

6.1644 

III 

B jerrum 

14 

2900 

5220 

49.20 

-5.9265 

III 

B jerrum 

15 

2945 

5301 

64.70 

-7.4047 

III 

B jerrum 

16 

3116 

5609 

76.10 

-8.6202 

III 

B jerrum 

Taking  0 = -13.1  in  the  above  equation  for  equi- 
librium we  have  the  following  calculated  values.  The  agreement 


betv/een  calculated  and  exp)erimental  results  can  be  seen  from 
fig.  (20)  page  128. 


Temp 

Eahr  abs  4.571  log,^Kp 
2300  24.146 

2500  19.925 

3000  11.861 

3500  6.134 


Temp 

Eahr  abs 
4000 
4500 
5000 
5500 


4.571  log,,  Xp 
1.855 
-1.462 
-4.116 
-6.297 


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Equilibri'um  eq.uation 


4.571 
K: 


Reaction 
+ i 0^ 

102100 


H^O 


- 6.2631  log„T+0. 0002361T+0. 033310'*  o!+C 


- _ y (g-y)'*'  / 1 

• p;:  ' (i-yV 


100  (l-y)  ^percent  of  dissociation  of  H^JD 
P^pressure  in  lbs.  per  sq,.  ft.  abs. 

T temperature  Pabr.  deg.  abs. 

Experimental  Data 

Pressure  =1  atm  = 2116.7  lb.  per  sq.  ft.  abs. 


Wo. 

Temperature 
C abs  P abs 

100  (l-y) 

4.571  log,^Z^ 

Method 

Investigator 

1 

1325 

2385 

0.0033 

23.2917 

IV 

Langmuir 

2 

1355 

2439 

0.0049 

22.6355 

IV 

IT 

3 

1393 

2507 

0.0068 

21.6458 

IV 

” (Yifartenberg 

4 

1397 

2515 

0.0078 

20.8400 

I 

Wernst  and  voh 

5 

1434 

2581 

0.0103 

20.4235 

IV 

Langmuir 

6 

1452 

2614 

0.0137 

19.5740 

IV 

tt 

7 

1458 

2624 

0.0147 

19.3647 

IV 

Tl 

8 

1474 

2653 

0.0140 

19.5096 

IV 

" (Wartenberg 

9 

1480 

2664 

0.0189 

18.6160 

I 

Wernst  and  von 

10 

1531 

2756 

0.0255 

17.7240 

IV 

Langmuir 

11 

1550 

2790 

0.0287 

17.3276 

IV 

" (V/artenberg 

12 

1561 

2810 

0.034 

16.8670 

I 

Wernst  and  von 

12 

1705 

3069 

0.102 

15.5950 

II 

L Owens te in 

14 

1783 

3209 

0.182 

11.8700 

II 

TT 

15 

1863 

3353 

0.354 

9.9320 

II 

TI 

16 

1968 

3542 

0.518 

8.7505 

II 

TT 

17 

2155 

3879 

1.180 

6.2900 

II 

von  Y/artenberg 

18 

2257 

4063 

1.770 

5.0271 

II 

IT  IT 

19 

2300 

4140 

2.600 

3.9608 

III 

Wernst 

20 

2642 

4756 

4.30 

2.3904 

III 

Bjerrum 

21 

2698 

4856 

7.50 

0.6820 

III 

IT 

22 

2761 

4970 

6. 60 

1.0776 

III 

Tl 

23 

2834 

5101 

9.80 

-0.1534 

III 

TT 

24 

2929 

5272 

11.1 

-0.5471 

III 

IT 

130 


Taking  C = 1.1  in  the  equilibrixim  equation  we  have 


the  following  calculated  values.  The  agreement  between  calcu- 
lated and  eiqjerimental  results  is  seen  from  Fig  (21)  page  131. 


Temp  Fahr  abs. 

4.571  log,. 

2500 

21.457 

3000 

14.365 

3500 

9.309 

4000 

5.542 

4500 

2.646 

5000 

0.366 

5500 

-1.455 

Heaction 


+ 00^5?  H^O  0 

By  subtracting  the  equilibrium  equation  for  the 
reaction  GO-*-^  CO,,  from  the  equilibrium  equation  for  the  re- 
action O^-tH^O  we  get  the  equilibrium  equation  for  the 

above  reaction  which  is 

4.571  log^,  -0.8750  log,^-1.0970'10'’T 0.1133  loVtl4. 2 

^ P..  • Pco.  “ xTi^T 


Experimental  Data 


Pressure  •=  1 atm  -2116.7  lb.  per  sq.  ft. 


No. 

Temperature 
C abs  P abs 

Log,.K^  4. 

571  log,,E:^ 

1 

1453 

2617 

2.39 

0.3784 

1.7276 

2 

1507 

2715 

2.46 

0.3909 

1.7868 

3 

1519 

2735 

2.62 

0.4183 

1.9121 

4 

1547 

2787 

2.31 

0.3636 

1.6620 

5 

1553 

2795 

2.25 

0.3522 

1.6099 

6 

1579 

2842 

2.69 

0.4299 

1.9651 

7 

1619 

2916 

3.31 

0.5198 

2.3760 

8 

1635 

2944 

3.10 

0.4914 

2.2462 

9 

1659 

2987 

3.59 

0.5551 

2.5374 

10 

1703 

3065 

3.13 

0.4955 

2.2649 

Method 

VI 


Investigator 
Aline r 


i 

I 


! 


V 

V. 


I , ..  • ;j 

‘i  ■ 


■f  - 


( ■ 


f 


/ 

a 


I 


\ '• . ! 


I 


1 

I 


j 

! 

t. 

t 

i 

I 


I 

t 


1 a 


0 : i'  ■.•.  J ,r  ■..  .:  {•  • 

.'J  0 ' ' J.  ■< ' V ' 


. • ' 'Tift'*.:  ij 

I 

' . n . ..  j , ; . ; 


% 


I 


I 


I 


A 


sv>v» 


!,  t j . 


f,“. 


A> 


o*‘f; 


I 4 


1 

J 


us’i^ 


v-yj*' 

o 


iv 


•\ 


a 

j' 


i«: 


\ 


<9 


c; 

(V 


t 


((■ 

(>■ 


it- 

ft- 


cT 

•r  - 
.V 


^ > ; > V* 

O r -9-  0 ^ 


g 

cP 

j 

\ 

> 

1 

; 

k : 

1 

X ‘ 

\ 

: "<^ 
\ 

c 


u 


C'’ 


0 

o 


:? 


CJ 

u? 


i'’  ■■■ 


'^L. 

,-^  ii} 

D-  ■> 

iP 

^ u> 


^2 

<P 


lD 
P (P 
><^ 

U/ 


C5’ 


J 


f,> 


C? 

<0 
U X-; 


P 


o 

o> 


) 

( 

^SCk 

. — ^ 

o. 

*Ti 

'i:  C.  \\  V'*  ‘ 

i ■ i • 

^ ^ .L- 

ip^j 

5/5 


132 


Bx^)erimental  Data  (cont.) 


Pressure  -1  atm  =2116.7  lb.  per  sq..  ft. 


Temperature 


Ho. 

C abs 

P abs 

Log,.K;^ 

4.571  log,,X^ 

Me thod 

11 

1706 

3071 

3.26 

0.5132 

2.3458  ^ 

VI 

12 

1747 

3145 

3.31 

0.5198 

2.3760 

13 

1798 

3238 

4.00 

0.6021 

2.7522 

14 

1503 

2705 

3.04 

0.4829 

2.2073 

VI 

15 

1528 

2752 

2.66 

0.4250 

1.9422 

16 

1538 

2770 

2.85 

0.4548 

2.0789 

17 

1578 

2840 

2.80 

0.4472 

2.0441 

18 

1586 

2856 

2.63 

0.4200 

1.9198 

19 

1597 

2875 

2.93 

0.4669 

2.1342 

20 

1643 

2959 

3.26 

0.5132 

2.3458 

21 

1763 

3174 

3.39 

0.5302 

2.4235 

22 

1773 

3192 

4.27 

0.6304 

2.8816 

23 

1783 

3210 

3.65 

0.5629 

2.5730 

24 

1795 

3231 

3.68 

0.5658 

2.5862 

25 

1798 

3237 

4.04 

0.6064 

2.7718 

26 

1824 

3282 

3.64 

0.5515 

2.5209 

959 

1726 

0.534 

-a2727 

-1.2465 

IV 

(■ 

28 

1059 

1906 

0.840 

-00757 

-0.3460 

29 

1159 

2086 

1.197 

0.0781 

0.3570 

30 

1259 

2266 

1.570 

0.1959 

0.8954 

31 

1278 

2300 

1.620 

0.2095 

0.9576 

32 

1359 

2446 

1.956 

0.2914 

1.3320 

33 

1478 

2660 

2.126 

0.3276 

1.4974 

34 

1678 

3020 

2.490 

0.3962 

1.8110 

35 

1031 

1856 

0.850 

-a0706 

-0.3227 

VII 

( 

36 

1111 

1920 

0.975 

-00110 

-0.0503 

37 

1134 

2041 

0.890 

-O.C506 

-0.2313 

38 

1227 

2209 

2.250 

0.3522 

1.6100 

39 

1283 

2309 

2.120 

0.3263 

1.4915 

40 

1333 

2399 

2.780 

0.4440 

2.0295 

Investigator 
Aline r 


Haber  & 
Hicbardt 


Halm 


Harries 


By  using  the  constant  of  integration  in  the  water 
gas  reaction  obtained  from  the  H^O  and  C 0^eq[uilibrium  equations 
we  get  the  agreement  shown  in  fig.  (22)  page  133  which  can  be 
considered  as  fairly  reasonable.  We  have  therefore  a double  ex- 
perimental basis  for  the  determination  of  the  constants  of  inte- 
gration in  our  equilibrium  equations  for  C 0^  and  H^O  dissociation. 

The  agreement  shown  between  the  calculated  and  ex- 


Fig22 

Fquf hbrn4m  of  the  Reaction  H;^0  + C0 

Showi ng  agreement  between  ca/cu lated 


134 


perimental  restilts  for  the  equilibrium  constants  may  be  con- 
sidered also  to  increase  the  probability  of  the  accuracy  of  the 
specific  heat  equations  used. 

I?e  action 


0 -h  Z E, 
51,850 


C H. 


4.571  log/^  - 19.7333  log  T^l.  21 -10’'^T-K).0293 '10'‘t7g 


r p*- 


P=pressure  lbs.  per  sq.  ft.  abs. 
T = temperature  degrees  Fahr.  abs. 


Experimental  Data 


Investigators  - Pring  and  Pairlie. 


Method  YII 

( b ) us  ing 

amorphous  carbon 

Temperature 

Ho. 

C abs. 

F abs 

4.571  log,, 

1 

1373 

2471 

-26.2129 

2 

1493 

2687 

-26.2855 

3 

1548 

2786 

-26.3774 

4 

1623 

2921 

-27.3137 

5 

1673 

3011 

-27.1358 

6 

1823 

3281 

-27.6615 

7 

1893 

3407 

-29.9960 

8 

1973 

3551 

-28.5527 

Method  YII 

( b ) using 

graphite  rod. 

Temperature 

Ho. 

C abs. 

F abs 

4.571  log„E:^ 

9 

1473 

2651 

-27.1439 

10 

1548 

2786 

-28.2491 

11 

1573 

2831 

-28.3096 

12 

1648 

2966 

-28.7588 

13 

1673 

3011 

-29.1065 

14 

1723 

3101 

-29.3750 

15 

1773 

3191 

-29.3956 

16 

1848 

3326 

-29.8478 

Method  YII  (b)  using  partly  graphitized  carbon  rod. 


Ho. 

Temperature 
C abs.  F abs. 

4.571  log„K_ 

17 

1573 

2831 

-27.7700 

18 

1623 

2921 

-27.7700 

135 


Method  VII  (b)  using  partly  graphitized  oarbon  rod.  (cont.) 


No. 

Tenperature 
C abs  P abs 

4.571  log,^  K 

19 

1723 

3101 

-27.6827 

20 

1748 

3146 

-28.9150 

21 

1813 

3263 

-29.2140 

Investigators  - Mayer  and  Altmayer. 

Method  VII  (a)  Passing  hydrogen  over  sugar  carbon  deposited  on 

nickle  or  cobalt  as  catalyser. 

Temperature 


No. 

C abs 

P abs 

4.571  log,,  EL 

1 

748 

1346 

-11.9020 

2 

780 

1404 

-13.5529 

3 

809 

1456 

-13.8736 

4 

840 

1512 

-14.3885 

5 

850 

1530 

-15.7218 

6 

880 

1584 

-16.3191 

7 

898 

1616 

-17.3384 

Investigators  - Clement  and  Adams. 

Method  VII  (a)  Passing  superheated  steam  over  coke. 


No. 

Temperature 
C abs  P abs 

4.571  log,,K 

1 

1173 

2110 

-26.4310 

2 

1273 

2290 

-26.5390 

3 

1373 

2470 

-29.5910 

4 

1473 

2650 

-29.9150 

5 

1573 

2830 

-29.0260 

Note:  In  cases  where  several  values  for  the  equilibrium  constant 
v/ere  given  by  the  investigator  for  any  one  temperature  the 
average  has  been  t^en  and  is  quoted  here. 


Putting  C=^£4.8  in  the  equilibrium  equation  we  get 
the  following  calculated  values: 


Temperature 
Pahr  abs 
1300 
1500 
1750 
2000 
2250 
2500 
2750 
3000 
3250 


4.571  log,.Zp 
-10.9843 
-15.1595 
-19.1267 
-22.1793 
-24.5915 
-26.5471 
-28.1577 
-29.5068 
-30.6473 


1 


136 


The  constant  places  the  calculated  curve 

through  the  Mayer  and  Altmayer  points  and  between  the  Clement  and 
Adams  and  Pring  and  Pairlie  points  as  shown  in  fig.  (23)  page  137. 
Consideiing  the  fact  that  the  specific  heat  eqLuation  used  for 
methane  is  based  on  low  temperature  measurements  only  the  agree- 
ment obtained  with  the  equilibrium  curve  is  very  good. 


Heaction  C+C  0^  2 C 0 

Equilibrium  equation  is  ^ l % 

4.571  log,JK:  ,-68.430  +9.1874  log  T - 1. 5844- lOT+0. 109*  10‘*T  f C 

^ " rn 

^ Poo 

^00-  ( ft 

P=  pressure  lb.  per  sq.  ft.  abs.  1 atmosphere  = 2116.7#  per  sq 

T = temperature  degrees  Pahr  abs. 

Experimental  Data 


Temperature 

llo. 

G abs 

P abs 

4.571  log,^E:^ 

Method 

Investigator 

1 

796 

1433 

4.0615 

VII  (a) 

Mayer  and  Jacoby 

2 

1861 

1550 

8.5347 

using 

3 

896 

1613 

9.8218 

sugar 

4 

940 

1692 

11.9337 

Cat  bon. 

5 

1043 

1877 

17.7770 

6 

1091 

1964 

19.3503 

7 

1073 

1931 

14.1329 

VII  (a) 

Clement  and  Adams 

8 

1123 

2021 

16.6544 

using 

9 

1173 

2111 

18.7188 

charcoal 

10 

1198 

2156 

19.6607 

11 

1273 

2291 

20.5041 

12 

1373 

2471 

22.1132 

13 

1173 

2111 

18.7995 

VII  (a) 

Clement  aid  Adams 

14 

1273 

2291 

19.0320 

us  ing 

15 

1373 

2471 

21.9055 

coke 

16 

1473 

2651 

23.7709 

17 

1573 

2831 

26.1465 

18 

923 

1661 

12.4445 

VII  (a) 

B oadouord 

19 

1073 

1931 

20.1926 

20 

1198 

2156 

21.4296 

21 

1073 

1931 

18.5127 

VII  (a) 

Rhea// and  V/heeler 

22 

1123 

2021 

20.4567 

23 

1173 

2111 

22.2460 

24 

1223 

2201 

23.4972 

25 

1273 

2291 

24.9294 

-•  i 


0 . 

-.'I  .fiv  UlO  01  a ij  i.'.  ''  . *► 

iiw’’  .j»w^  ‘if, 


(-'■ 

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■ '‘t'15 

«• 

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. . o:  • 


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cl'k'Xl 

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t: 

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kr 

xsi 

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S . ‘ 


ngZ3 

Egud/bnum  of  the  Reaction  Cti^ 

5how/ng  agreement  between  ca/culated 
and  expenmentat  values. 


..  I 


o 

O 


V 

7 

ff 

:3i 

t 


if 


5- 

? 

r 


Jr 

I ^ 

,; 

u > 

O » 4- 


a 


fi 


^ ^ V' 

tp  u 

fy  ^ 'T 

2 ip  O' 
u? 

^ t^* 

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■f 


138 


Bscperimental  Data  (cont) 


Eo. 

Temperature 
0 abs  F abs 

4.571  log,^K^  Method 

Investigator 

26 

1323 

2381 

26.1162 

She ad  and  Wheeler 

27 

1373 

2471 

27.5758 

28 

1473 

2651 

29.9264 

Taking  0*26.4  we  get  the  following  calculated  values; 


Temperature 


Fahr  abs 

4.571  log,* 

1500 

7.2275 

1750 

13.9589 

2000 

18.9799 

2250 

22.8715 

2500 

25.9767 

2750 

28  .4822 

3000 

30.5638 

The  curve  as  shown  in  fig.  (24)  page  139  passes 
through  the  Bondourad  value  at  T=2156  F (ahs).  At  the  lower  end 
the  curve  passes  between  the  Bondourad  point  and  the  Mayer  and 
Jacoby  points  while  at  the  upper  end  it  passes  between  the  Bhead 
and  YvTieeler  and  the  Clement  and  Adams  points.  The  agreement  be- 
tween calculated  and  experiments!,  values  may  be  td^ien  as  satis- 
factory. 


Reaction 

C + 20^  C0j,^+2  H^O 

We  obtain  the  ecjiilibrium  equation  for  this  reac- 
tion indirectly  from  the  equilibrium  equations  of  the  follovdng 
reactions : 

200-^0^-^  20 
2 + 0^  -^2  E^O 

^ 2Q  Q 

CHy  ^ 0-hzE^ 


Adding 


0 20,.  C0,_-^2  Hj,0 


i' 


IV 


ui  J-  ':  L^i  i.v  f.oi,)i  '<i 


i 

I 

:i 

} ' ■ -V  ' i\ 

' i 

i;  _ . 


• Hkf  .I'r*  1 >1  f 


V.  i#  •#  ./»vV  WW  • U 

'>c'T  ' O'  . ; Lv,-  t.  .:,j  t.:  w.-oaoL'"' 


L.  . 


ffO  a O 


..u;-  C • • ' ..to;  , p L 


. ":f  . , 


;'i  f 


r.>'>  i 


{)  i:>  c' 

•.* ; y*  .r. 


'■^1 


page  139^ 

M ~ 


y ^ 

■f  J" 
'V 


c 

c 


S .y 


s' 

J 

tf' 

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a' 

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7* 

if 

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r' 

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$. 

<v 


V 

L> 

tr. 


/ i 


I . I 


1 i 


I 


140 


Adding  the  equilihrium  equations  for  these  reactions 
we  have  for  the  equilihriura  equation  for  the  oomhustion  of 
methane  with  oxygen. 

'x/17  npn  -9  ■*- 

4.571  log^^I£p=£^AA^ -^5. 6183 -log,^T -0.0552  *10  T-0. 0133*10  T-22.4 
a- 

TT  P • P 


He action 
+ 0^  2 il  0 

Assuming  that  the  specific  heat  of  F 0 is  the  same 
as  that  of  and  0^  at  all  temperr  tures , and  using  Thomsen’s 
value  for  the  heat  of  reaction  we  have  for  the  equilibrium 


equation: 


4.571 


-77400 

T 


~h  4.06 


K. 


■/V^ 

The  constant  4.86  is  based  on  the  experimentil  data 


obtained  by  ITernst  using  the  streaming  method.  The  results  are 
given  in  the  following  table : 


Temperature 
C abs  F abs 

io  NO 

4.5V1  log„|A^ 

-77400 

— — 

Const. 

1811 

3260 

78.92 

20.72 

0.37 

-18.592 

-23.742 

5.150 

1877 

3379 

78.89 

20.69 

0.42 

-16.130 

-22.906 

4.776 

2033 

3659 

78.78 

20.58 

0.64 

-16.406 

-21.153 

4.747 

2195 

3951 

78.61 

20.42 

0.97 

-14.738 

-19.590 

4.852 

2580 

4644 

78.08 

19.88 

2.05 

-11.736 

-16.667 

4.931 

2675 

4815 

77.98 

19.78 

2.23 

-11.389 

-16.074 

4.685 

ave.  4.858 


141 


ScL'ulli'briiiin  liquations 

4.571  log,^  - 6.2631  log,^  T +0.2361 'lo'^fO. 033?- 10^ Ttl.l 

CO  +10^  00^ 

4.571  log,^  IC^  = - 5.3881  log, + 1.3335 -lO^T-0. 08 -lo'S-lS.l 


+ co^  n^o  +C0 
4.571  log^^K^ri^ 


- 0.8750  log,^  T-  1.0970 -lO^T+0. 1133 -1  o' V+14^ 


G + 2 ^ CH^ 

4.571  log/„K^,^l|^  -19.7333  log,^  T+1. 21-10r-K).029 •10~^tV24. 8 


C + C0^->2  CO 

4.571  10S,^K^-T-^^^*-  + 9.1874  I T -1.9844  l6*^T+ 0.109 -lo'^T+26.4 
C H^  + 20  -+00^+  2-H^O 

4.571  log,^  +5.6183  log,„  T-0.0552-l6^T-0.ol33 -loV-22.4 

K^tO^  2 ITO 

4.571  +4.86 


•V 


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142 


References  for  Experiment-:.!  Data 

1.  Eernst  and  von  Wartenberg 

2.  Langmuir 

3.  Lov/enstein 
4»  Bmich 

5.  Bjerrum 


on  Reaction  CO 

Zeit  Phys  Chem  5G-548  (1906) 

Jour  Am  Chem  Soc  28-1357  (1906) 

Zeit  Phys  Chem  54-707  (1906) 

llonatshefte  f Chem  26-1011  (1905) 
Zeit  Phys  Chem  79-357  (1906) 


References  for  Experimental  Data  on  equilibrium  in 

Reaction  t Hj,0 


1.  Langmuir 

2.  Lowenstein 

3.  von  Y/artenberg 

4.  Rernst 

5.  Herns t and  von  Wartenberg 

6.  Bjerrum 

7.  Seigel 


Jour  Am  Chem  Soc 
Zeit  Phys  Chem 
Zeit  Phys  Chem 
Zeit  AnprgChem 
Zeit  Phys  Chem 
Zeit  Phys  Chem 
Zeit  Phys  Chem 


28-1357  (1906) 
54-  715  (1906) 
56-  513  (1906) 
45-  130  (1905) 
56-  534  (1906) 
79-  513  (1912) 
87-  641  (1914) 


References  for  Experimental  Data  on  equilibrium  in 

Reaction  E^-h  CO^^-t  H^O  GO 

1.  Allner  Jour  f Gasbel  48  (1905) 

pp  1035,  1057,  1081,  1107. 

2.  Haber  and  Richardt  Zeit  Anorg  Chem  38-  5 (1904) 

3.  Hahn  Zeit  Phys  Chem  42-  705  (1902-5) 

4.  Harries  Jour  f Gasbel  37-  82  (1894) 

References  for  Equilibrium  Data  on  Reaction  C f-2  C H^ 

1.  Pring  and  Pairlie  Jour  Chem  Soc  101-  91  (1912) 

2.  Mayer  and  Altmeyer  Berlin  Ber  40-2135  (1907) 

3.  Clement  aiidAdams  U S Bureau  of  Bui  # 7 

Mines  p 41  (1911) 


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143 


5eferenoes  for  I^eaotion 
1.  Mayer  and  Jacoby 
£.  Clement  and  Adams 

3.  Boudourad 

4.  Hhead  and  Vi/heeler 


0 i-C  0^  £ C 0 

Jour  f Gasbel 
U.S.  Bureau  of  Mines 
Compt  Bendus 
Jour  Chem  Soc 


5£  (1909)  £8£,  305 
Bui  # 7 (1909) 
130rl3£  (1900) 

97  (1910)  2176 

89  (1911)  1140 


References  for  Equilibrium  Data  for  Reaction  0),^  -^  £ IT  0 
1.  Nemst  Zeit  f Anorg  Chem  49  £13  (1906) 


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APPENDIX  D 


DETSmilMTION  OP  THE  IIAXD.TUH 
POSSIBLE  PEHCENTAGE  OP  IT  0 PRESENT  AT 
THE  POINT  OP  LIAXIMUH  TEIIPERilTURE  IN  THE  GAS  ENGINE 


« ’ 
aasar  naSBiTv^*  sa 


■ *J 

>\mf. 


.'lalMritt'' 


144 


Appendix  D 

To  determine  the  maximum  percent  of  IT  0 present  at  the 
point  of  maximum  temperature  in  the  gas  engine. 


The  equilibrium  equation  is 


4.571  log 


- P 


•77400 

T 

(A) 


-y-  4. 86 


In  order  to  malLe  the  partial  pressure  of  the  IT  0 
as  large  as  possible  the  partial  pressures  of  il^and  0^  must  be 
as  large  as  possible  for  a given  value  of 

Assuming  a gaseous  mixture  with  50^  excess  air  and 
a dissociation  of  10^  at  the  point  of  maximum  temperature  the  IT^ 
and  0 ^ will  be  present  in  the  ratio  of  9.5  volumes  of  IT^  to 
1 vol.  of  0^  . 

Let  V = of  IT  ^transformed  to  IT  0 

Then  at  equilibrium  point  the  mixture  is 

mols 

IT  0 = 2 X 9.5  Y * 19  Y 

=»  9.5  (1-Y)  = 9.5  - 9.5  Y 

0;^  - 1-9.5  Y = 1 - 9.5  Y 

M = Total  number  of  mols  present 


P , iLI.p  p 9.5  (1-Y)  .p 


i\/o 


p 


1-9.5  Y 



Substituting  the  expressions  for  partial  pressure  in  equation  (A) 
v/e  have 

(19  Y)’'  =■  9.5  (l-Y)-(l-9.5  Y)  ' 

Solving  for  Y 

Y 


-10.5  72.25 


76  - 19 

r“ 


T^ing  T =5000  P (abs)  a liberal  maximum  temperature  IT  ^0.00477. 


145 

Solving  equation  (B) 

V -0.0064 

That  is  0.64^  of  the  nitrogen  present  is  transformed  into  IT  0 
under  the  most  favorable  conditions.  Decreasing  the  maximum  tem- 
perature and  the  amount  of  excess  air  will  both  decrease  the 
value  of  Y. 

The  accuracy  of  the  physical  and  chemical  constants 
used  in  this  discussion  does  not  warrant  the  consideration  of  a 
quantity  which  amounts  to  less  than  0.64$b  of  the  total  gas  volume 
present.  ITo  appreciable  error  will  be  introduced  into  the  calcu- 
lations of  the  equilibrium  conditions  by  neglecting  the  reaction 

11^  t 0^  2 IT  0 . 


